:: FDIFF_1 semantic presentation
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theorem Th1: :: FDIFF_1:1
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:: deftheorem Def1 defines convergent_to_0 FDIFF_1:def 1 :
reconsider cs = NAT --> 0 as Real_Sequence by FUNCOP_1:57;
Lm1:
for n being Nat holds cs . n = 0
by FUNCOP_1:13;
:: deftheorem FDIFF_1:def 2 :
canceled;
:: deftheorem Def3 defines REST-like FDIFF_1:def 3 :
reconsider cf = REAL --> 0 as Function of REAL , REAL by FUNCOP_1:57;
Lm2:
for r being Real holds cf . r = 0
by FUNCOP_1:13;
:: deftheorem Def4 defines linear FDIFF_1:def 4 :
theorem :: FDIFF_1:2
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canceled;
theorem :: FDIFF_1:3
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canceled;
theorem :: FDIFF_1:4
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canceled;
theorem :: FDIFF_1:5
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canceled;
theorem Th6: :: FDIFF_1:6
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theorem Th7: :: FDIFF_1:7
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theorem Th8: :: FDIFF_1:8
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theorem Th9: :: FDIFF_1:9
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theorem Th10: :: FDIFF_1:10
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theorem Th11: :: FDIFF_1:11
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:: deftheorem Def5 defines is_differentiable_in FDIFF_1:def 5 :
:: deftheorem Def6 defines diff FDIFF_1:def 6 :
:: deftheorem Def7 defines is_differentiable_on FDIFF_1:def 7 :
theorem :: FDIFF_1:12
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canceled;
theorem :: FDIFF_1:13
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canceled;
theorem :: FDIFF_1:14
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canceled;
theorem Th15: :: FDIFF_1:15
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theorem Th16: :: FDIFF_1:16
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theorem :: FDIFF_1:17
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:: deftheorem Def8 defines `| FDIFF_1:def 8 :
theorem :: FDIFF_1:18
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canceled;
theorem :: FDIFF_1:19
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theorem Th20: :: FDIFF_1:20
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theorem Th21: :: FDIFF_1:21
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theorem Th22: :: FDIFF_1:22
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theorem Th23: :: FDIFF_1:23
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theorem Th24: :: FDIFF_1:24
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theorem :: FDIFF_1:25
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theorem :: FDIFF_1:26
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theorem :: FDIFF_1:27
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theorem :: FDIFF_1:28
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theorem :: FDIFF_1:29
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theorem :: FDIFF_1:30
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theorem :: FDIFF_1:31
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theorem Th32: :: FDIFF_1:32
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theorem :: FDIFF_1:33
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theorem :: FDIFF_1:34
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theorem :: FDIFF_1:35
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