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

proof

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

end;

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

proof

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

end;

then ( sec | B is with_symmetrical_domain & sec | B is quasi_even ) by A2;
hence sec is_even_on B by A1; :: thesis: verum