defpred S1[ object , object ] means ( ( P1[\$1] implies \$2 = F3(\$1) ) & ( P2[\$1] implies \$2 = F4(\$1) ) );
defpred S2[ object ] means ( P1[\$1] or P2[\$1] );
consider A being set such that
A4: for x being object holds
( x in A iff ( x in F1() & S2[x] ) ) from A5: for x being object st x in A holds
ex y being object st S1[x,y]
proof
let x be object ; :: thesis: ( x in A implies ex y being object st S1[x,y] )
assume A6: x in A ; :: thesis: ex y being object st S1[x,y]
then A7: x in F1() by A4;
now :: thesis: ex y being set ex y being object st S1[x,y]
per cases ( P1[x] or P2[x] ) by A4, A6;
suppose A8: P1[x] ; :: thesis: ex y being set ex y being object st S1[x,y]
take y = F3(x); :: thesis: ex y being object st S1[x,y]
P2[x] by A1, A7, A8;
hence ex y being object st S1[x,y] ; :: thesis: verum
end;
suppose A9: P2[x] ; :: thesis: ex y being set ex y being object st S1[x,y]
take y = F4(x); :: thesis: ex y being object st S1[x,y]
P1[x] by A1, A7, A9;
hence ex y being object st S1[x,y] ; :: thesis: verum
end;
end;
end;
hence ex y being object st S1[x,y] ; :: thesis: verum
end;
consider f being Function such that
A10: ( dom f = A & ( for x being object st x in A holds
S1[x,f . x] ) ) from A11: A c= F1() by A4;
rng f c= F2()
proof
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in rng f or x in F2() )
assume x in rng f ; :: thesis: x in F2()
then consider y being object such that
A12: y in dom f and
A13: x = f . y by FUNCT_1:def 3;
now :: thesis: x in F2()end;
hence x in F2() ; :: thesis: verum
end;
then reconsider f = f as PartFunc of F1(),F2() by ;
take f ; :: thesis: ( ( for x being object holds
( x in dom f iff ( x in F1() & ( P1[x] or P2[x] ) ) ) ) & ( for x being object st x in dom f holds
( ( P1[x] implies f . x = F3(x) ) & ( P2[x] implies f . x = F4(x) ) ) ) )

thus for x being object holds
( x in dom f iff ( x in F1() & ( P1[x] or P2[x] ) ) ) by ; :: thesis: for x being object st x in dom f holds
( ( P1[x] implies f . x = F3(x) ) & ( P2[x] implies f . x = F4(x) ) )

let x be object ; :: thesis: ( x in dom f implies ( ( P1[x] implies f . x = F3(x) ) & ( P2[x] implies f . x = F4(x) ) ) )
assume x in dom f ; :: thesis: ( ( P1[x] implies f . x = F3(x) ) & ( P2[x] implies f . x = F4(x) ) )
hence ( ( P1[x] implies f . x = F3(x) ) & ( P2[x] implies f . x = F4(x) ) ) by A10; :: thesis: verum