begin
theorem Th1:
:: deftheorem Def1 defines convergent_to_0 FDIFF_1:def 1 :
reconsider cs = NAT --> 0 as Real_Sequence by FUNCOP_1:57;
:: 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;
:: deftheorem Def4 defines linear FDIFF_1:def 4 :
theorem
canceled;
theorem
canceled;
theorem
canceled;
theorem
canceled;
theorem Th6:
theorem Th7:
theorem Th8:
theorem Th9:
theorem Th10:
theorem Th11:
:: 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
canceled;
theorem
canceled;
theorem
canceled;
theorem Th15:
theorem Th16:
theorem
:: deftheorem Def8 defines `| FDIFF_1:def 8 :
theorem
canceled;
theorem
theorem Th20:
theorem Th21:
theorem Th22:
theorem Th23:
theorem Th24:
theorem Th25:
theorem
theorem
theorem
theorem
theorem
theorem
theorem Th32:
theorem
theorem Th34:
theorem
:: deftheorem Def9 defines differentiable FDIFF_1:def 9 :
Lm1:
{} REAL is closed
Lm2:
[#] REAL is open
theorem