defpred S1[ object , object ] means for x1, y1, w1 being object st \$1 = [x1,y1,w1] holds
P1[x1,y1,w1,\$2];
A2: for x being object st x in [:F1(),F2(),F3():] holds
ex z being object st
( z in F4() & S1[x,z] )
proof
let x be object ; :: thesis: ( x in [:F1(),F2(),F3():] implies ex z being object st
( z in F4() & S1[x,z] ) )

assume x in [:F1(),F2(),F3():] ; :: thesis: ex z being object st
( z in F4() & S1[x,z] )

then consider x1, y1, w1 being object such that
A3: ( x1 in F1() & y1 in F2() & w1 in F3() ) and
A4: x = [x1,y1,w1] by MCART_1:68;
consider z being object such that
A5: z in F4() and
A6: P1[x1,y1,w1,z] by A1, A3;
take z ; :: thesis: ( z in F4() & S1[x,z] )
thus z in F4() by A5; :: thesis: S1[x,z]
let x2, y2, w2 be object ; :: thesis: ( x = [x2,y2,w2] implies P1[x2,y2,w2,z] )
assume A7a: x = [x2,y2,w2] ; :: thesis: P1[x2,y2,w2,z]
then ( [x1,y1] = [x2,y2] & w1 = w2 ) by ;
then ( x1 = x2 & y1 = y2 ) by XTUPLE_0:1;
hence P1[x2,y2,w2,z] by A6, A7a, A4, XTUPLE_0:1; :: thesis: verum
end;
consider f being Function of [:F1(),F2(),F3():],F4() such that
A8: for x being object st x in [:F1(),F2(),F3():] holds
S1[x,f . x] from take f ; :: thesis: for x, y, w being object st x in F1() & y in F2() & w in F3() holds
P1[x,y,w,f . (x,y,w)]

thus for x, y, w being object st x in F1() & y in F2() & w in F3() holds
P1[x,y,w,f . (x,y,w)] by ; :: thesis: verum