let r be Real; :: thesis: for A being symmetrical Subset of COMPLEX
for F being PartFunc of REAL ,REAL st F is_even_on A holds
F - r is_even_on A
let A be symmetrical Subset of COMPLEX ; :: thesis: for F being PartFunc of REAL ,REAL st F is_even_on A holds
F - r is_even_on A
let F be PartFunc of REAL ,REAL ; :: thesis: ( F is_even_on A implies F - r is_even_on A )
assume F is_even_on A ; :: thesis: F - r is_even_on A
then A1: ( A c= dom F & F | A is even ) by Def5;
A2: A c= dom (F - r) by A1, VALUED_1:3;
A3: dom ((F - r) | A) = A by RELAT_1:91, A2;
for x being Real st x in dom ((F - r) | A) & - x in dom ((F - r) | A) holds
((F - r) | A) . (- x) = ((F - r) | A) . x

proof

let x be Real; :: thesis: ( x in dom ((F - r) | A) & - x in dom ((F - r) | A) implies ((F - r) | A) . (- x) = ((F - r) | A) . x )
assume A5: ( x in dom ((F - r) | A) & - x in dom ((F - r) | A) ) ; :: thesis: ((F - r) | A) . (- x) = ((F - r) | A) . x
then A6: ( x in dom (F | A) & - x in dom (F | A) ) by RELAT_1:91, A1, A3;
((F - r) | A) . (- x) = ((F - r) | A) /. (- x) by PARTFUN1:def 8, A5
.= (F - r) /. (- x) by PARTFUN2:35, A2, A3, A5
.= (F - r) . (- x) by PARTFUN1:def 8, A2, A3, A5
.= (F . (- x)) - r by VALUED_1:3, A1, A3, A5
.= (F /. (- x)) - r by PARTFUN1:def 8, A3, A1, A5
.= ((F | A) /. (- x)) - r by PARTFUN2:35, A1, A3, A5
.= ((F | A) . (- x)) - r by PARTFUN1:def 8, A6
.= ((F | A) . x) - r by A1, Def3, A6
.= ((F | A) /. x) - r by PARTFUN1:def 8, A6
.= (F /. x) - r by PARTFUN2:35, A1, A3, A5
.= (F . x) - r by PARTFUN1:def 8, A1, A3, A5
.= (F - r) . x by VALUED_1:3, A1, A3, A5
.= (F - r) /. x by PARTFUN1:def 8, A2, A3, A5
.= ((F - r) | A) /. x by PARTFUN2:35, A2, A3, A5
.= ((F - r) | A) . x by PARTFUN1:def 8, A5 ;
hence ((F - r) | A) . (- x) = ((F - r) | A) . x ; :: thesis: verum

end;

then ( (F - r) | A is with_symmetrical_domain & (F - r) | A is quasi_even ) by Def3, A3, Def2;
hence F - r is_even_on A by A2, Def5; :: thesis: verum