for th being Real st th in dom (cosec | ].0,(PI / 2).]) holds
cosec | ].0,(PI / 2).] is_continuous_in th
proof
let th be
Real;
( th in dom (cosec | ].0,(PI / 2).]) implies cosec | ].0,(PI / 2).] is_continuous_in th )
A1:
sin is_differentiable_in th
by SIN_COS:64;
assume A2:
th in dom (cosec | ].0,(PI / 2).])
;
cosec | ].0,(PI / 2).] is_continuous_in th
then
th in ].0,(PI / 2).]
by RELAT_1:57;
then
sin . th <> 0
by Lm4, COMPTRIG:7;
then A3:
cosec is_continuous_in th
by A1, FCONT_1:10, FDIFF_1:24;
now for rseq being Real_Sequence st rng rseq c= dom (cosec | ].0,(PI / 2).]) & rseq is convergent & lim rseq = th holds
( (cosec | ].0,(PI / 2).]) /* rseq is convergent & (cosec | ].0,(PI / 2).]) . th = lim ((cosec | ].0,(PI / 2).]) /* rseq) )let rseq be
Real_Sequence;
( rng rseq c= dom (cosec | ].0,(PI / 2).]) & rseq is convergent & lim rseq = th implies ( (cosec | ].0,(PI / 2).]) /* rseq is convergent & (cosec | ].0,(PI / 2).]) . th = lim ((cosec | ].0,(PI / 2).]) /* rseq) ) )assume that A4:
rng rseq c= dom (cosec | ].0,(PI / 2).])
and A5:
(
rseq is
convergent &
lim rseq = th )
;
( (cosec | ].0,(PI / 2).]) /* rseq is convergent & (cosec | ].0,(PI / 2).]) . th = lim ((cosec | ].0,(PI / 2).]) /* rseq) )A6:
dom (cosec | ].0,(PI / 2).]) = ].0,(PI / 2).]
by Th4, RELAT_1:62;
now for n being Element of NAT holds ((cosec | ].0,(PI / 2).]) /* rseq) . n = (cosec /* rseq) . nlet n be
Element of
NAT ;
((cosec | ].0,(PI / 2).]) /* rseq) . n = (cosec /* rseq) . n
dom rseq = NAT
by SEQ_1:1;
then
rseq . n in rng rseq
by FUNCT_1:def 3;
then A7:
(cosec | ].0,(PI / 2).]) . (rseq . n) = cosec . (rseq . n)
by A4, A6, FUNCT_1:49;
(cosec | ].0,(PI / 2).]) . (rseq . n) = ((cosec | ].0,(PI / 2).]) /* rseq) . n
by A4, FUNCT_2:108;
hence
((cosec | ].0,(PI / 2).]) /* rseq) . n = (cosec /* rseq) . n
by A4, A6, A7, Th4, FUNCT_2:108, XBOOLE_1:1;
verum end; then A8:
(cosec | ].0,(PI / 2).]) /* rseq = cosec /* rseq
by FUNCT_2:63;
A9:
rng rseq c= dom cosec
by A4, A6, Th4;
then
cosec . th = lim (cosec /* rseq)
by A3, A5, FCONT_1:def 1;
hence
(
(cosec | ].0,(PI / 2).]) /* rseq is
convergent &
(cosec | ].0,(PI / 2).]) . th = lim ((cosec | ].0,(PI / 2).]) /* rseq) )
by A2, A3, A5, A9, A8, Lm36, FCONT_1:def 1;
verum end;
hence
cosec | ].0,(PI / 2).] is_continuous_in th
by FCONT_1:def 1;
verum
end;
hence
cosec | ].0,(PI / 2).] is continuous
by FCONT_1:def 2; verum