now :: thesis: for x being object st x in { z where z is Element of E : for b being Element of E st b in B holds
z - b in A } holds
x in the carrier of E

let x be object ; :: thesis: ( x in { z where z is Element of E : for b being Element of E st b in B holds
z - b in A } implies x in the carrier of E )
assume x in { z where z is Element of E : for b being Element of E st b in B holds
z - b in A } ; :: thesis: x in the carrier of E
then ex z being Element of E st
( x = z & ( for b being Element of E st b in B holds
z - b in A ) ) ;
hence x in the carrier of E ; :: thesis: verum

end;

hence { z where z is Element of E : for b being Element of E st b in B holds
z - b in A } is binary-image of E by TARSKI:def 3; :: thesis: verum