let B be symmetrical Subset of REAL; :: thesis: ( B c= dom cosec implies cosec is_odd_on B )
assume A1: B c= dom cosec ; :: thesis: cosec is_odd_on B
then A2: dom (cosec | B) = B by RELAT_1:62;
A3: for x being Real st x in B holds
cosec . (- x) = - (cosec . x)

proof

let x be Real; :: thesis: ( x in B implies cosec . (- x) = - (cosec . x) )
assume A4: x in B ; :: thesis: cosec . (- x) = - (cosec . x)
then - x in B by Def1;
then cosec . (- x) = (sin . (- x)) " by A1, RFUNCT_1:def 2
.= (- (sin . x)) " by SIN_COS:30
.= - ((sin . x) ") by XCMPLX_1:222
.= - (cosec . x) by A1, A4, RFUNCT_1:def 2 ;
hence cosec . (- x) = - (cosec . x) ; :: thesis: verum

end;

for x being Real st x in dom (cosec | B) & - x in dom (cosec | B) holds
(cosec | B) . (- x) = - ((cosec | B) . x)

proof

let x be Real; :: thesis: ( x in dom (cosec | B) & - x in dom (cosec | B) implies (cosec | B) . (- x) = - ((cosec | B) . x) )
assume that
A5: x in dom (cosec | B) and
A6: - x in dom (cosec | B) ; :: thesis: (cosec | B) . (- x) = - ((cosec | B) . x)
(cosec | B) . (- x) = (cosec | B) /. (- x) by A6, PARTFUN1:def 6
.= cosec /. (- x) by A1, A2, A6, PARTFUN2:17
.= cosec . (- x) by A1, A6, PARTFUN1:def 6
.= - (cosec . x) by A3, A5
.= - (cosec /. x) by A1, A5, PARTFUN1:def 6
.= - ((cosec | B) /. x) by A1, A2, A5, PARTFUN2:17
.= - ((cosec | B) . x) by A5, PARTFUN1:def 6 ;
hence (cosec | B) . (- x) = - ((cosec | B) . x) ; :: thesis: verum

end;

then ( cosec | B is with_symmetrical_domain & cosec | B is quasi_odd ) by A2;
hence cosec is_odd_on B by A1; :: thesis: verum