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