let F1, F2 be FUNCTION_DOMAIN of the carrier of G, the carrier of G; :: thesis: ( ( for f being Element of Funcs ( the carrier of G, the carrier of G) holds
( f in F1 iff ex a being Element of G st
for x being Element of G holds f . x = x |^ a ) ) & ( for f being Element of Funcs ( the carrier of G, the carrier of G) holds
( f in F2 iff ex a being Element of G st
for x being Element of G holds f . x = x |^ a ) ) implies F1 = F2 )
assume that
A4: for f being Element of Funcs ( the carrier of G, the carrier of G) holds
( f in F1 iff ex a being Element of G st
for x being Element of G holds f . x = x |^ a ) and
A5: for f being Element of Funcs ( the carrier of G, the carrier of G) holds
( f in F2 iff ex a being Element of G st
for x being Element of G holds f . x = x |^ a ) ; :: thesis: F1 = F2
A6: F2 c= F1

proof

let q be object ; :: according to TARSKI:def 3 :: thesis: ( not q in F2 or q in F1 )
assume A7: q in F2 ; :: thesis: q in F1
then q is Function of the carrier of G, the carrier of G by FUNCT_2:def 12;
then reconsider b1 = q as Element of Funcs ( the carrier of G, the carrier of G) by FUNCT_2:9;
ex a being Element of G st
for x being Element of G holds b1 . x = x |^ a by A5, A7;
hence q in F1 by A4; :: thesis: verum

end;

F1 c= F2

proof

let q be object ; :: according to TARSKI:def 3 :: thesis: ( not q in F1 or q in F2 )
assume A8: q in F1 ; :: thesis: q in F2
then q is Function of the carrier of G, the carrier of G by FUNCT_2:def 12;
then reconsider b1 = q as Element of Funcs ( the carrier of G, the carrier of G) by FUNCT_2:9;
ex a being Element of G st
for x being Element of G holds b1 . x = x |^ a by A4, A8;
hence q in F2 by A5; :: thesis: verum

end;

hence F1 = F2 by A6, XBOOLE_0:def 10; :: thesis: verum