defpred S₁[ set , set ] means ex o being object of st
( $1 = o & $2 = (t2 ! o) * (t1 ! o) );
A2: for a being Element of ex j being set st S₁[a,j]

proof

let a be Element of ; :: thesis: ex j being set st S₁[a,j]
reconsider o = a as object of ;
ex j being set st j = (t2 ! o) * (t1 ! o) ;
hence ex j being set st S₁[a,j] ; :: thesis: verum

end;

consider t being ManySortedSet of the carrier of A such that
A3: for a being Element of holds S₁[a,t . a] from PBOOLE:sch 6(A2);
A4: F is_transformable_to F2 by A1, Th4;
for a being object of holds t . a is Morphism of ,

proof

let o be Element of ; :: thesis: t . o is Morphism of ,
ex a being object of st
( o = a & t . o = (t2 ! a) * (t1 ! a) ) by A3;
hence t . o is Morphism of , ; :: thesis: verum

end;

then reconsider t' = t as transformation of F,F2 by A4, Def2;
take t' ; :: thesis: for a being object of holds t' ! a = (t2 ! a) * (t1 ! a)
let a be Element of ; :: thesis: t' ! a = (t2 ! a) * (t1 ! a)
ex o being object of st
( a = o & t . a = (t2 ! o) * (t1 ! o) ) by A3;
hence t' ! a = (t2 ! a) * (t1 ! a) by A4, Def4; :: thesis: verum