let A be symmetrical Subset of COMPLEX; :: thesis: for F, G being PartFunc of REAL,REAL st F is_odd_on A & G is_odd_on A holds
F (#) G is_even_on A
let F, G be PartFunc of REAL,REAL; :: thesis: ( F is_odd_on A & G is_odd_on A implies F (#) G is_even_on A )
assume that
A1: F is_odd_on A and
A2: G is_odd_on A ; :: thesis: F (#) G is_even_on A
A3: A c= dom G by A2;
A4: G | A is odd by A2;
A5: A c= dom F by A1;
then A6: A c= (dom F) /\ (dom G) by A3, XBOOLE_1:19;
A7: (dom F) /\ (dom G) = dom (F (#) G) by VALUED_1:def 4;
then A8: dom ((F (#) G) | A) = A by A5, A3, RELAT_1:62, XBOOLE_1:19;
A9: F | A is odd by A1;
for x being Real st x in dom ((F (#) G) | A) & - x in dom ((F (#) G) | A) holds
((F (#) G) | A) . (- x) = ((F (#) G) | A) . x

proof

let x be Real; :: thesis: ( x in dom ((F (#) G) | A) & - x in dom ((F (#) G) | A) implies ((F (#) G) | A) . (- x) = ((F (#) G) | A) . x )
assume that
A10: x in dom ((F (#) G) | A) and
A11: - x in dom ((F (#) G) | A) ; :: thesis: ((F (#) G) | A) . (- x) = ((F (#) G) | A) . x
A12: x in dom (F | A) by A5, A8, A10, RELAT_1:62;
A13: x in dom (G | A) by A3, A8, A10, RELAT_1:62;
A14: - x in dom (F | A) by A5, A8, A11, RELAT_1:62;
A15: - x in dom (G | A) by A3, A8, A11, RELAT_1:62;
reconsider x = x as Element of REAL by XREAL_0:def 1;
((F (#) G) | A) . (- x) = ((F (#) G) | A) /. (- x) by A11, PARTFUN1:def 6
.= (F (#) G) /. (- x) by A6, A7, A8, A11, PARTFUN2:17
.= (F (#) G) . (- x) by A6, A7, A11, PARTFUN1:def 6
.= (F . (- x)) * (G . (- x)) by A6, A7, A11, VALUED_1:def 4
.= (F /. (- x)) * (G . (- x)) by A5, A11, PARTFUN1:def 6
.= (F /. (- x)) * (G /. (- x)) by A3, A11, PARTFUN1:def 6
.= ((F | A) /. (- x)) * (G /. (- x)) by A5, A8, A11, PARTFUN2:17
.= ((F | A) /. (- x)) * ((G | A) /. (- x)) by A3, A8, A11, PARTFUN2:17
.= ((F | A) . (- x)) * ((G | A) /. (- x)) by A14, PARTFUN1:def 6
.= ((F | A) . (- x)) * ((G | A) . (- x)) by A15, PARTFUN1:def 6
.= (- ((F | A) . x)) * ((G | A) . (- x)) by A9, A12, A14, Def6
.= (- ((F | A) . x)) * (- ((G | A) . x)) by A4, A13, A15, Def6
.= ((F | A) . x) * ((G | A) . x)
.= ((F | A) /. x) * ((G | A) . x) by A12, PARTFUN1:def 6
.= ((F | A) /. x) * ((G | A) /. x) by A13, PARTFUN1:def 6
.= (F /. x) * ((G | A) /. x) by A5, A8, A10, PARTFUN2:17
.= (F /. x) * (G /. x) by A3, A8, A10, PARTFUN2:17
.= (F . x) * (G /. x) by A5, A10, PARTFUN1:def 6
.= (F . x) * (G . x) by A3, A10, PARTFUN1:def 6
.= (F (#) G) . x by A6, A7, A10, VALUED_1:def 4
.= (F (#) G) /. x by A6, A7, A10, PARTFUN1:def 6
.= ((F (#) G) | A) /. x by A6, A7, A8, A10, PARTFUN2:17
.= ((F (#) G) | A) . x by A10, PARTFUN1:def 6 ;
hence ((F (#) G) | A) . (- x) = ((F (#) G) | A) . x ; :: thesis: verum

end;

then ( (F (#) G) | A is with_symmetrical_domain & (F (#) G) | A is quasi_even ) by A8;
hence F (#) G is_even_on A by A6, A7; :: thesis: verum