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