defpred S₁[ object , object ] means ex A being binary-image of E st
( $1 = A & $2 = A (+) B );

A1: now :: thesis: for x being object st x in bool the carrier of E holds
ex y being object st
( y in bool the carrier of E & S₁[x,y] )

let x be object ; :: thesis: ( x in bool the carrier of E implies ex y being object st
( y in bool the carrier of E & S₁[x,y] ) )
assume x in bool the carrier of E ; :: thesis: ex y being object st
( y in bool the carrier of E & S₁[x,y] )
then reconsider A = x as binary-image of E ;
set y = A (+) B;
A (+) B c= the carrier of E ;
hence ex y being object st
( y in bool the carrier of E & S₁[x,y] ) ; :: thesis: verum

end;

consider f being Function of (bool the carrier of E),(bool the carrier of E) such that
A2: for x being object st x in bool the carrier of E holds
S₁[x,f . x] from FUNCT_2:sch 1(A1);
take f ; :: thesis: for A being binary-image of E holds f . A = A (+) B

now :: thesis: for A being binary-image of E holds f . A = A (+) B

let A be binary-image of E; :: thesis: f . A = A (+) B
ex X being binary-image of E st
( A = X & f . A = X (+) B ) by A2;
hence f . A = A (+) B ; :: thesis: verum

end;

hence for A being binary-image of E holds f . A = A (+) B ; :: thesis: verum