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