for th being Real st th in dom (sec | [.0,(PI / 2).[) holds
sec | [.0,(PI / 2).[ is_continuous_in th

proof

let th be Real; :: thesis: ( th in dom (sec | [.0,(PI / 2).[) implies sec | [.0,(PI / 2).[ is_continuous_in th )
A1: cos is_differentiable_in th by SIN_COS:63;
assume A2: th in dom (sec | [.0,(PI / 2).[) ; :: thesis: sec | [.0,(PI / 2).[ is_continuous_in th
then th in [.0,(PI / 2).[ by RELAT_1:57;
then cos . th <> 0 by Lm1, COMPTRIG:11;
then A3: sec is_continuous_in th by A1, FCONT_1:10, FDIFF_1:24;

now :: thesis: for rseq being Real_Sequence st rng rseq c= dom (sec | [.0,(PI / 2).[) & rseq is convergent & lim rseq = th holds
( (sec | [.0,(PI / 2).[) /* rseq is convergent & (sec | [.0,(PI / 2).[) . th = lim ((sec | [.0,(PI / 2).[) /* rseq) )

let rseq be Real_Sequence; :: thesis: ( rng rseq c= dom (sec | [.0,(PI / 2).[) & rseq is convergent & lim rseq = th implies ( (sec | [.0,(PI / 2).[) /* rseq is convergent & (sec | [.0,(PI / 2).[) . th = lim ((sec | [.0,(PI / 2).[) /* rseq) ) )
assume that
A4: rng rseq c= dom (sec | [.0,(PI / 2).[) and
A5: ( rseq is convergent & lim rseq = th ) ; :: thesis: ( (sec | [.0,(PI / 2).[) /* rseq is convergent & (sec | [.0,(PI / 2).[) . th = lim ((sec | [.0,(PI / 2).[) /* rseq) )
A6: dom (sec | [.0,(PI / 2).[) = [.0,(PI / 2).[ by Th1, RELAT_1:62;

now :: thesis: for n being Element of NAT holds ((sec | [.0,(PI / 2).[) /* rseq) . n = (sec /* rseq) . n

let n be Element of NAT ; :: thesis: ((sec | [.0,(PI / 2).[) /* rseq) . n = (sec /* rseq) . n
dom rseq = NAT by SEQ_1:1;
then rseq . n in rng rseq by FUNCT_1:def 3;
then A7: (sec | [.0,(PI / 2).[) . (rseq . n) = sec . (rseq . n) by A4, A6, FUNCT_1:49;
(sec | [.0,(PI / 2).[) . (rseq . n) = ((sec | [.0,(PI / 2).[) /* rseq) . n by A4, FUNCT_2:108;
hence ((sec | [.0,(PI / 2).[) /* rseq) . n = (sec /* rseq) . n by A4, A6, A7, Th1, FUNCT_2:108, XBOOLE_1:1; :: thesis: verum

end;

then A8: (sec | [.0,(PI / 2).[) /* rseq = sec /* rseq by FUNCT_2:63;
A9: rng rseq c= dom sec by A4, A6, Th1;
then sec . th = lim (sec /* rseq) by A3, A5, FCONT_1:def 1;
hence ( (sec | [.0,(PI / 2).[) /* rseq is convergent & (sec | [.0,(PI / 2).[) . th = lim ((sec | [.0,(PI / 2).[) /* rseq) ) by A2, A3, A5, A9, A8, Lm33, FCONT_1:def 1; :: thesis: verum

end;

hence sec | [.0,(PI / 2).[ is_continuous_in th by FCONT_1:def 1; :: thesis: verum

end;

hence sec | [.0,(PI / 2).[ is continuous by FCONT_1:def 2; :: thesis: verum