Show the Proof

import proveit
# Automation is not needed when only showing a stored proof:
proveit.defaults.automation = False # This will speed things up.
proveit.defaults.inline_pngs = False # Makes files smaller.
%show_proof

	step type	requirements	statement
0	instantiation	1, 2, 3	⊢
	: , : , :
1	reference	45	⊢
2	instantiation	295, 4, 5	⊢
	: , : , :
3	instantiation	25, 6, 7, 8	⊢
	: , : , :
4	instantiation	295, 9, 10	⊢
	: , : , :
5	instantiation	336, 11	⊢
	: , : , :
6	instantiation	12, 139, 380	⊢
	: , :
7	instantiation	13, 76, 90	⊢
	:
8	instantiation	418, 389, 14	⊢
	: , : , :
9	instantiation	45, 15, 16	⊢
	: , : , :
10	instantiation	282, 420, 417, 353, 17, 354, 351, 18, 19, 20*	⊢
	: , : , : , : , : , :
11	instantiation	338, 21, 22	⊢
	: , : , :
12	theorem		⊢
	proveit.numbers.exponentiation.exp_complex_nonzero_closure
13	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.nonzero_complex_is_complex_nonzero
14	instantiation	238, 23, 164	⊢
	: , :
15	instantiation	295, 24, 62	⊢
	: , : , :
16	instantiation	25, 26, 351, 27*	⊢
	: , : , :
17	instantiation	369	⊢
	: , :
18	instantiation	418, 389, 28	⊢
	: , : , :
19	instantiation	418, 389, 29	⊢
	: , : , :
20	instantiation	295, 30, 31	⊢
	: , : , :
21	instantiation	280, 353, 417, 420, 354, 32, 380, 142, 351	⊢
	: , : , : , : , : , :
22	instantiation	338, 33, 34	⊢
	: , : , :
23	instantiation	418, 408, 35	⊢
	: , : , :
24	instantiation	45, 36, 37	⊢
	: , : , :
25	theorem		⊢
	proveit.numbers.division.frac_cancel_left
26	instantiation	418, 377, 122	⊢
	: , : , :
27	instantiation	338, 38, 39	⊢
	: , : , :
28	instantiation	395, 72	⊢
	:
29	instantiation	395, 73	⊢
	:
30	instantiation	255, 40	⊢
	: , :
31	instantiation	336, 41	⊢
	: , : , :
32	instantiation	369	⊢
	: , :
33	instantiation	231, 420, 353, 354, 380, 142, 351	⊢
	: , : , : , : , : , : , :
34	instantiation	282, 353, 417, 420, 354, 42, 380, 351, 142, 43*	⊢
	: , : , : , : , : , :
35	instantiation	418, 415, 44	⊢
	: , : , :
36	instantiation	45, 46, 47	⊢
	: , : , :
37	instantiation	48, 380, 49, 414	⊢
	: , : , :
38	instantiation	50, 417, 51, 52, 53, 54	⊢
	: , : , : , :
39	instantiation	336, 55	⊢
	: , : , :
40	instantiation	56, 57, 58	⊢
	: , :
41	instantiation	255, 59	⊢
	: , :
42	instantiation	369	⊢
	: , :
43	theorem		⊢
	proveit.numbers.numerals.decimals.add_2_1
44	instantiation	418, 419, 60	⊢
	: , : , :
45	theorem		⊢
	proveit.logic.equality.sub_left_side_into
46	instantiation	295, 61, 62	⊢
	: , : , :
47	instantiation	240, 63, 64, 65	⊢
	: , : , : , :
48	theorem		⊢
	proveit.numbers.exponentiation.posnat_power_of_product
49	instantiation	418, 389, 193	⊢
	: , : , :
50	axiom		⊢
	proveit.core_expr_types.operations.operands_substitution
51	instantiation	369	⊢
	: , :
52	instantiation	369	⊢
	: , :
53	instantiation	66, 106, 351	⊢
	: , :
54	instantiation	67, 106, 412, 68*, 305*	⊢
	: , : , :
55	instantiation	240, 69, 70, 71	⊢
	: , : , : , :
56	theorem		⊢
	proveit.numbers.negation.distribute_neg_through_binary_sum
57	instantiation	418, 389, 72	⊢
	: , : , :
58	instantiation	418, 389, 73	⊢
	: , : , :
59	instantiation	74, 414, 75, 106, 351, 76, 90	⊢
	: , : , :
60	theorem		⊢
	proveit.numbers.numerals.decimals.nat3
61	instantiation	215, 77, 78	⊢
	: , : , :
62	instantiation	79, 420, 417, 353, 80, 354, 380, 351, 233, 81*, 82*	⊢
	: , : , : , : , : , :
63	instantiation	83, 414, 380	⊢
	: , :
64	instantiation	84, 417, 85, 86, 87	⊢
	: , : , : , :
65	theorem		⊢
	proveit.numbers.numerals.decimals.mult_2_2
66	theorem		⊢
	proveit.numbers.multiplication.commutation
67	theorem		⊢
	proveit.numbers.exponentiation.product_of_posnat_powers
68	instantiation	379, 106	⊢
	:
69	instantiation	338, 88, 89	⊢
	: , : , :
70	instantiation	291	⊢
	:
71	instantiation	255, 103	⊢
	: , :
72	instantiation	180, 121, 91, 90	⊢
	: , :
73	instantiation	180, 399, 91, 90	⊢
	: , :
74	theorem		⊢
	proveit.numbers.division.distribute_frac_through_sum
75	instantiation	369	⊢
	: , :
76	instantiation	418, 389, 91	⊢
	: , : , :
77	instantiation	95, 417, 353, 92, 354, 399, 93, 94	⊢
	: , : , : , : , : , :
78	instantiation	95, 420, 399, 96, 97	⊢
	: , : , : , : , : , :
79	theorem		⊢
	proveit.numbers.multiplication.distribute_through_sum
80	instantiation	369	⊢
	: , :
81	instantiation	98, 380	⊢
	:
82	instantiation	295, 99, 100	⊢
	: , : , :
83	theorem		⊢
	proveit.numbers.exponentiation.exp_nat_pos_expansion
84	theorem		⊢
	proveit.core_expr_types.operations.operands_substitution_via_tuple
85	instantiation	101, 417	⊢
	: , :
86	instantiation	369	⊢
	: , :
87	instantiation	102	⊢
	:
88	instantiation	336, 103	⊢
	: , : , :
89	instantiation	190, 104	⊢
	:
90	instantiation	105, 106, 380, 107	⊢
	: , :
91	instantiation	170, 121, 417	⊢
	: , :
92	instantiation	369	⊢
	: , :
93	instantiation	395, 113	⊢
	:
94	instantiation	111, 110, 108, 109	⊢
	: , :
95	theorem		⊢
	proveit.numbers.addition.weak_bound_via_term_bound
96	instantiation	395, 110	⊢
	:
97	instantiation	111, 112, 113, 114	⊢
	: , :
98	theorem		⊢
	proveit.numbers.multiplication.elim_one_right
99	instantiation	295, 115, 116	⊢
	: , : , :
100	instantiation	338, 117, 118	⊢
	: , : , :
101	theorem		⊢
	proveit.core_expr_types.tuples.range_from1_len_typical_eq
102	theorem		⊢
	proveit.numbers.numerals.decimals.reduce_2_repeats
103	instantiation	336, 119	⊢
	: , : , :
104	instantiation	120, 380, 357	⊢
	: , :
105	theorem		⊢
	proveit.numbers.exponentiation.exp_not_eq_zero
106	instantiation	418, 389, 121	⊢
	: , : , :
107	instantiation	324, 122	⊢
	:
108	instantiation	180, 399, 123, 124	⊢
	: , :
109	instantiation	133, 134, 166, 125, 126	⊢
	: , : , :
110	instantiation	180, 399, 127, 128	⊢
	: , :
111	theorem		⊢
	proveit.numbers.negation.negated_weak_bound
112	instantiation	180, 399, 129, 130	⊢
	: , :
113	instantiation	180, 399, 131, 132	⊢
	: , :
114	instantiation	133, 134, 174, 135, 136	⊢
	: , : , :
115	instantiation	137, 351, 367, 138, 139	⊢
	: , : , : , : , :
116	instantiation	336, 140	⊢
	: , : , :
117	instantiation	336, 141	⊢
	: , : , :
118	instantiation	190, 142	⊢
	:
119	instantiation	313, 351, 372, 348, 143*	⊢
	: , :
120	theorem		⊢
	proveit.numbers.addition.add_complex_closure_bin
121	instantiation	238, 390, 164	⊢
	: , :
122	instantiation	418, 334, 144	⊢
	: , : , :
123	instantiation	152, 149, 146	⊢
	: , :
124	instantiation	324, 145	⊢
	:
125	instantiation	418, 405, 171	⊢
	: , : , :
126	instantiation	158, 149, 146, 150, 147, 148	⊢
	: , : , :
127	instantiation	152, 149, 150	⊢
	: , :
128	instantiation	153, 151	⊢
	:
129	instantiation	152, 390, 247	⊢
	: , :
130	instantiation	153, 154	⊢
	:
131	instantiation	418, 381, 174	⊢
	: , : , :
132	instantiation	324, 155	⊢
	:
133	theorem		⊢
	proveit.numbers.division.weak_div_from_denom_bound__all_pos
134	instantiation	418, 156, 157	⊢
	: , : , :
135	instantiation	418, 405, 173	⊢
	: , : , :
136	instantiation	158, 390, 193, 247, 185, 159	⊢
	: , : , :
137	theorem		⊢
	proveit.numbers.division.mult_frac_cancel_numer_left
138	instantiation	418, 377, 160	⊢
	: , : , :
139	instantiation	418, 377, 161	⊢
	: , : , :
140	instantiation	338, 162, 163	⊢
	: , : , :
141	instantiation	179, 351	⊢
	:
142	instantiation	418, 389, 164	⊢
	: , : , :
143	instantiation	190, 357	⊢
	:
144	instantiation	341, 393, 165	⊢
	: , :
145	instantiation	418, 334, 166	⊢
	: , : , :
146	instantiation	170, 193, 417	⊢
	: , :
147	instantiation	167, 390, 193, 247, 168, 249	⊢
	: , : , :
148	instantiation	176, 200	⊢
	:
149	instantiation	418, 408, 169	⊢
	: , : , :
150	instantiation	170, 247, 417	⊢
	: , :
151	instantiation	418, 172, 171	⊢
	: , : , :
152	theorem		⊢
	proveit.numbers.multiplication.mult_real_closure_bin
153	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nonzero_if_in_rational_nonzero
154	instantiation	418, 172, 173	⊢
	: , : , :
155	instantiation	418, 334, 174	⊢
	: , : , :
156	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_nonneg_within_real_nonneg
157	instantiation	418, 175, 420	⊢
	: , : , :
158	theorem		⊢
	proveit.numbers.multiplication.weak_bound_via_right_factor_bound
159	instantiation	176, 417	⊢
	:
160	instantiation	418, 387, 177	⊢
	: , : , :
161	instantiation	418, 334, 375	⊢
	: , : , :
162	instantiation	336, 178	⊢
	: , : , :
163	instantiation	179, 380	⊢
	:
164	instantiation	180, 399, 385, 348	⊢
	: , :
165	instantiation	391, 392, 375, 348	⊢
	: , :
166	instantiation	358, 181, 182	⊢
	: , :
167	theorem		⊢
	proveit.numbers.exponentiation.exp_pos_lesseq
168	instantiation	183, 184, 185	⊢
	: , :
169	instantiation	418, 415, 186	⊢
	: , : , :
170	theorem		⊢
	proveit.numbers.exponentiation.exp_real_closure_nat_power
171	instantiation	188, 191, 187	⊢
	: , :
172	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.rational_pos_within_rational_nonzero
173	instantiation	188, 406, 202	⊢
	: , :
174	instantiation	358, 393, 214	⊢
	: , :
175	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nat_within_rational_nonneg
176	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.natural_lower_bound
177	instantiation	418, 401, 189	⊢
	: , : , :
178	instantiation	190, 380	⊢
	:
179	theorem		⊢
	proveit.numbers.division.frac_one_denom
180	theorem		⊢
	proveit.numbers.division.div_real_closure
181	instantiation	418, 405, 191	⊢
	: , : , :
182	instantiation	192, 193, 194	⊢
	:
183	theorem		⊢
	proveit.logic.booleans.conjunction.and_if_both
184	instantiation	195, 196	⊢
	:
185	instantiation	225, 383, 197, 294, 198, 199*	⊢
	: , : , :
186	instantiation	418, 419, 200	⊢
	: , : , :
187	instantiation	201, 202, 411	⊢
	: , :
188	theorem		⊢
	proveit.numbers.multiplication.mult_rational_pos_closure_bin
189	instantiation	418, 410, 412	⊢
	: , : , :
190	theorem		⊢
	proveit.numbers.multiplication.elim_one_left
191	instantiation	418, 413, 203	⊢
	: , : , :
192	theorem		⊢
	proveit.numbers.exponentiation.sqrd_pos_closure
193	instantiation	238, 399, 244	⊢
	: , :
194	instantiation	324, 204	⊢
	:
195	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nonneg_if_in_real_nonneg
196	instantiation	418, 205, 214	⊢
	: , : , :
197	instantiation	418, 381, 286	⊢
	: , : , :
198	instantiation	206, 390, 272, 207, 361, 208, 209*	⊢
	: , : , :
199	instantiation	338, 210, 211	⊢
	: , : , :
200	theorem		⊢
	proveit.numbers.numerals.decimals.nat4
201	theorem		⊢
	proveit.numbers.exponentiation.exp_rational_pos_closure
202	instantiation	212, 256, 213	⊢
	:
203	theorem		⊢
	proveit.numbers.numerals.decimals.posnat4
204	instantiation	418, 334, 214	⊢
	: , : , :
205	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real_nonneg
206	theorem		⊢
	proveit.numbers.exponentiation.exp_monotonicity_large_base_less_eq
207	instantiation	238, 228, 226	⊢
	: , :
208	instantiation	215, 216, 217	⊢
	: , : , :
209	instantiation	218, 393, 286, 330	⊢
	: , :
210	instantiation	280, 353, 417, 420, 354, 219, 380, 233, 373	⊢
	: , : , : , : , : , :
211	instantiation	338, 220, 221	⊢
	: , : , :
212	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.pos_rational_is_rational_pos
213	instantiation	222, 223	⊢
	: , :
214	instantiation	341, 392, 298	⊢
	: , :
215	theorem		⊢
	proveit.numbers.ordering.transitivity_less_eq_less_eq
216	instantiation	224, 272	⊢
	:
217	instantiation	225, 226, 227, 228, 229, 230*	⊢
	: , : , :
218	theorem		⊢
	proveit.numbers.exponentiation.exponent_log_with_same_base
219	instantiation	369	⊢
	: , :
220	instantiation	231, 420, 353, 354, 380, 233, 373	⊢
	: , : , : , : , : , : , :
221	instantiation	282, 353, 417, 420, 354, 232, 380, 373, 233, 296*	⊢
	: , : , : , : , : , :
222	theorem		⊢
	proveit.numbers.addition.subtraction.pos_difference
223	instantiation	234, 383, 390, 235, 236, 296*, 237*	⊢
	: , : , :
224	theorem		⊢
	proveit.numbers.rounding.ceil_x_ge_x
225	theorem		⊢
	proveit.numbers.addition.weak_bound_via_left_term_bound
226	instantiation	395, 300	⊢
	:
227	instantiation	238, 300, 239	⊢
	: , :
228	instantiation	308, 309, 344	⊢
	: , : , :
229	axiom		⊢
	proveit.physics.quantum.QPE._t_req
230	instantiation	240, 241, 242, 243	⊢
	: , : , : , :
231	theorem		⊢
	proveit.numbers.addition.leftward_commutation
232	instantiation	369	⊢
	: , :
233	instantiation	418, 389, 244	⊢
	: , : , :
234	theorem		⊢
	proveit.numbers.addition.strong_bound_via_left_term_bound
235	instantiation	418, 408, 245	⊢
	: , : , :
236	instantiation	246, 390, 247, 248, 249	⊢
	: , : , :
237	instantiation	338, 250, 251	⊢
	: , : , :
238	theorem		⊢
	proveit.numbers.addition.add_real_closure_bin
239	instantiation	418, 408, 252	⊢
	: , : , :
240	theorem		⊢
	proveit.logic.equality.four_chain_transitivity
241	instantiation	338, 253, 254	⊢
	: , : , :
242	instantiation	291	⊢
	:
243	instantiation	255, 273	⊢
	: , :
244	instantiation	418, 381, 298	⊢
	: , : , :
245	instantiation	265, 256, 403	⊢
	: , :
246	theorem		⊢
	proveit.numbers.ordering.less_eq_add_right_strong
247	instantiation	418, 408, 256	⊢
	: , : , :
248	theorem		⊢
	proveit.physics.quantum.QPE._e_value_ge_two
249	instantiation	257, 414	⊢
	:
250	instantiation	336, 258	⊢
	: , : , :
251	instantiation	338, 259, 260	⊢
	: , : , :
252	instantiation	418, 415, 261	⊢
	: , : , :
253	instantiation	336, 262	⊢
	: , : , :
254	instantiation	338, 263, 264	⊢
	: , : , :
255	theorem		⊢
	proveit.logic.equality.equals_reversal
256	instantiation	265, 303, 266	⊢
	: , :
257	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.natural_pos_is_pos
258	instantiation	338, 267, 268	⊢
	: , : , :
259	instantiation	280, 353, 417, 420, 354, 269, 284, 351, 373	⊢
	: , : , : , : , : , :
260	instantiation	270, 351, 284, 271	⊢
	: , : , :
261	instantiation	319, 272	⊢
	:
262	instantiation	336, 273	⊢
	: , : , :
263	instantiation	280, 353, 417, 420, 354, 274, 289, 277, 275	⊢
	: , : , : , : , : , :
264	instantiation	276, 289, 277, 278	⊢
	: , : , :
265	theorem		⊢
	proveit.numbers.addition.add_rational_closure_bin
266	instantiation	418, 415, 279	⊢
	: , : , :
267	instantiation	280, 353, 417, 420, 354, 281, 284, 373, 380	⊢
	: , : , : , : , : , :
268	instantiation	282, 420, 417, 353, 283, 354, 284, 373, 380, 285*	⊢
	: , : , : , : , : , :
269	instantiation	369	⊢
	: , :
270	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_23
271	instantiation	291	⊢
	:
272	instantiation	328, 393, 286, 330	⊢
	: , :
273	instantiation	336, 287	⊢
	: , : , :
274	instantiation	369	⊢
	: , :
275	instantiation	288, 289	⊢
	:
276	theorem		⊢
	proveit.numbers.addition.subtraction.add_cancel_triple_13
277	instantiation	418, 389, 290	⊢
	: , : , :
278	instantiation	291	⊢
	:
279	instantiation	418, 292, 293	⊢
	: , : , :
280	theorem		⊢
	proveit.numbers.addition.disassociation
281	instantiation	369	⊢
	: , :
282	theorem		⊢
	proveit.numbers.addition.association
283	instantiation	369	⊢
	: , :
284	instantiation	418, 389, 294	⊢
	: , : , :
285	instantiation	295, 296, 297	⊢
	: , : , :
286	instantiation	341, 393, 298	⊢
	: , :
287	instantiation	336, 299	⊢
	: , : , :
288	theorem		⊢
	proveit.numbers.negation.complex_closure
289	instantiation	418, 389, 300	⊢
	: , : , :
290	instantiation	418, 408, 301	⊢
	: , : , :
291	axiom		⊢
	proveit.logic.equality.equals_reflexivity
292	theorem		⊢
	proveit.numbers.number_sets.integers.neg_int_within_int
293	instantiation	302, 412	⊢
	:
294	instantiation	418, 408, 303	⊢
	: , : , :
295	theorem		⊢
	proveit.logic.equality.sub_right_side_into
296	instantiation	304, 351, 380, 305	⊢
	: , : , :
297	instantiation	306, 380, 373	⊢
	: , :
298	instantiation	391, 392, 335, 315	⊢
	: , :
299	instantiation	336, 307	⊢
	: , : , :
300	instantiation	308, 309, 363	⊢
	: , : , :
301	instantiation	418, 415, 310	⊢
	: , : , :
302	theorem		⊢
	proveit.numbers.negation.int_neg_closure
303	instantiation	418, 311, 312	⊢
	: , : , :
304	theorem		⊢
	proveit.numbers.addition.subtraction.subtract_from_add
305	theorem		⊢
	proveit.numbers.numerals.decimals.add_1_1
306	theorem		⊢
	proveit.numbers.addition.commutation
307	instantiation	313, 351, 314, 315, 316*	⊢
	: , :
308	theorem		⊢
	proveit.logic.sets.inclusion.unfold_subset_eq
309	instantiation	317, 318	⊢
	: , :
310	instantiation	319, 320	⊢
	:
311	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.rational_nonzero_within_rational
312	instantiation	321, 388, 322	⊢
	: , :
313	theorem		⊢
	proveit.numbers.division.div_as_mult
314	instantiation	418, 389, 323	⊢
	: , : , :
315	instantiation	324, 325	⊢
	:
316	instantiation	338, 326, 327	⊢
	: , : , :
317	theorem		⊢
	proveit.logic.sets.inclusion.relax_proper_subset
318	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nat_pos_within_real
319	axiom		⊢
	proveit.numbers.rounding.ceil_is_an_int
320	instantiation	328, 393, 329, 330	⊢
	: , :
321	theorem		⊢
	proveit.numbers.exponentiation.exp_rational_nonzero_closure
322	instantiation	331, 332, 333	⊢
	: , :
323	instantiation	418, 381, 335	⊢
	: , : , :
324	theorem		⊢
	proveit.numbers.number_sets.real_numbers.nonzero_if_in_real_nonzero
325	instantiation	418, 334, 335	⊢
	: , : , :
326	instantiation	336, 337	⊢
	: , : , :
327	instantiation	338, 339, 340	⊢
	: , : , :
328	theorem		⊢
	proveit.numbers.logarithms.log_real_pos_real_closure
329	instantiation	341, 393, 342	⊢
	: , :
330	instantiation	364, 343	⊢
	: , :
331	theorem		⊢
	proveit.numbers.addition.add_int_closure_bin
332	instantiation	418, 362, 344	⊢
	: , : , :
333	instantiation	345, 346	⊢
	:
334	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real_nonzero
335	instantiation	358, 393, 375	⊢
	: , :
336	axiom		⊢
	proveit.logic.equality.substitution
337	instantiation	347, 380, 372, 383, 394, 348, 349*	⊢
	: , : , :
338	axiom		⊢
	proveit.logic.equality.equals_transitivity
339	instantiation	350, 420, 417, 353, 355, 354, 351, 356, 357	⊢
	: , : , : , : , : , :
340	instantiation	352, 353, 417, 354, 355, 356, 357	⊢
	: , : , : , :
341	theorem		⊢
	proveit.numbers.addition.add_real_pos_closure_bin
342	instantiation	358, 382, 359	⊢
	: , :
343	instantiation	360, 420, 417, 361	⊢
	: , :
344	axiom		⊢
	proveit.physics.quantum.QPE._t_in_natural_pos
345	theorem		⊢
	proveit.numbers.negation.int_closure
346	instantiation	418, 362, 363	⊢
	: , : , :
347	theorem		⊢
	proveit.numbers.exponentiation.real_power_of_product
348	instantiation	364, 365	⊢
	: , :
349	instantiation	366, 367, 412, 368*	⊢
	: , :
350	theorem		⊢
	proveit.numbers.multiplication.disassociation
351	instantiation	418, 389, 399	⊢
	: , : , :
352	theorem		⊢
	proveit.numbers.multiplication.elim_one_any
353	axiom		⊢
	proveit.numbers.number_sets.natural_numbers.zero_in_nats
354	theorem		⊢
	proveit.core_expr_types.tuples.tuple_len_0_typical_eq
355	instantiation	369	⊢
	: , :
356	instantiation	418, 389, 370	⊢
	: , : , :
357	instantiation	371, 372, 373	⊢
	: , :
358	theorem		⊢
	proveit.numbers.multiplication.mult_real_pos_closure_bin
359	instantiation	374, 375, 383	⊢
	: , :
360	theorem		⊢
	proveit.numbers.ordering.less_is_not_eq_nat
361	theorem		⊢
	proveit.numbers.numerals.decimals.less_1_2
362	theorem		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_int
363	axiom		⊢
	proveit.physics.quantum.QPE._n_in_natural_pos
364	theorem		⊢
	proveit.logic.equality.not_equals_symmetry
365	instantiation	376, 398, 385, 386	⊢
	: , :
366	theorem		⊢
	proveit.numbers.exponentiation.neg_power_as_div
367	instantiation	418, 377, 378	⊢
	: , : , :
368	instantiation	379, 380	⊢
	:
369	theorem		⊢
	proveit.numbers.numerals.decimals.tuple_len_2_typical_eq
370	instantiation	418, 381, 382	⊢
	: , : , :
371	theorem		⊢
	proveit.numbers.exponentiation.exp_complex_closure
372	instantiation	418, 389, 385	⊢
	: , : , :
373	instantiation	418, 389, 383	⊢
	: , : , :
374	theorem		⊢
	proveit.numbers.exponentiation.exp_real_pos_closure
375	instantiation	384, 385, 386	⊢
	:
376	theorem		⊢
	proveit.numbers.ordering.less_is_not_eq
377	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_nonzero_within_complex_nonzero
378	instantiation	418, 387, 388	⊢
	: , : , :
379	theorem		⊢
	proveit.numbers.exponentiation.complex_x_to_first_power_is_x
380	instantiation	418, 389, 390	⊢
	: , : , :
381	theorem		⊢
	proveit.numbers.number_sets.real_numbers.real_pos_within_real
382	instantiation	391, 392, 393, 394	⊢
	: , :
383	instantiation	395, 399	⊢
	:
384	theorem		⊢
	proveit.numbers.number_sets.real_numbers.pos_real_is_real_pos
385	instantiation	396, 398, 399, 400	⊢
	: , : , :
386	instantiation	397, 398, 399, 400	⊢
	: , : , :
387	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_nonzero_within_real_nonzero
388	instantiation	418, 401, 402	⊢
	: , : , :
389	theorem		⊢
	proveit.numbers.number_sets.complex_numbers.real_within_complex
390	instantiation	418, 408, 403	⊢
	: , : , :
391	theorem		⊢
	proveit.numbers.division.div_real_pos_closure
392	instantiation	418, 405, 404	⊢
	: , : , :
393	instantiation	418, 405, 406	⊢
	: , : , :
394	instantiation	407, 414	⊢
	:
395	theorem		⊢
	proveit.numbers.negation.real_closure
396	theorem		⊢
	proveit.numbers.number_sets.real_numbers.all_in_interval_oc__is__real
397	theorem		⊢
	proveit.numbers.number_sets.real_numbers.interval_oc_lower_bound
398	theorem		⊢
	proveit.numbers.number_sets.real_numbers.zero_is_real
399	instantiation	418, 408, 409	⊢
	: , : , :
400	axiom		⊢
	proveit.physics.quantum.QPE._eps_in_interval
401	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nonzero_int_within_rational_nonzero
402	instantiation	418, 410, 414	⊢
	: , : , :
403	instantiation	418, 415, 411	⊢
	: , : , :
404	instantiation	418, 413, 412	⊢
	: , : , :
405	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_pos_within_real_pos
406	instantiation	418, 413, 414	⊢
	: , : , :
407	theorem		⊢
	proveit.numbers.number_sets.natural_numbers.nonzero_if_is_nat_pos
408	theorem		⊢
	proveit.numbers.number_sets.real_numbers.rational_within_real
409	instantiation	418, 415, 416	⊢
	: , : , :
410	theorem		⊢
	proveit.numbers.number_sets.integers.nat_pos_within_nonzero_int
411	instantiation	418, 419, 417	⊢
	: , : , :
412	theorem		⊢
	proveit.numbers.numerals.decimals.posnat1
413	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.nat_pos_within_rational_pos
414	theorem		⊢
	proveit.numbers.numerals.decimals.posnat2
415	theorem		⊢
	proveit.numbers.number_sets.rational_numbers.int_within_rational
416	instantiation	418, 419, 420	⊢
	: , : , :
417	theorem		⊢
	proveit.numbers.numerals.decimals.nat2
418	theorem		⊢
	proveit.logic.sets.inclusion.superset_membership_from_proper_subset
419	theorem		⊢
	proveit.numbers.number_sets.integers.nat_within_int
420	theorem		⊢
	proveit.numbers.numerals.decimals.nat1
*equality replacement requirements