defpred S₁[ set , set ] means ex o being object of A st
( $1 = o & $2 in <^(F1 . o),(F2 . o)^> );
A2: for a being Element of A ex j being set st S₁[a,j]

proof

let a be Element of A; :: thesis: ex j being set st S₁[a,j]
reconsider o = a as object of A ;
<^(F1 . o),(F2 . o)^> <> {} by A1, Def1;
then consider j being set such that
A3: j in <^(F1 . o),(F2 . o)^> by XBOOLE_0:def 1;
thus ex j being set st S₁[a,j] by A3; :: thesis: verum

end;

consider t being ManySortedSet of such that
A4: for a being Element of A holds S₁[a,t . a] from PBOOLE:sch 6(A2);
take t ; :: thesis: for a being object of A holds t . a is Morphism of (F1 . a),(F2 . a)
let a be object of A; :: thesis: t . a is Morphism of (F1 . a),(F2 . a)
ex o being object of A st
( a = o & t . a in <^(F1 . o),(F2 . o)^> ) by A4;
hence t . a is Morphism of (F1 . a),(F2 . a) ; :: thesis: verum