reserve V, C, x, a, b for set;
reserve A, B for Element of SubstitutionSet (V, C);

theorem
  A ^ B = A implies for a be set st a in A ex b be set st b in B & b c= a
proof
  assume
A1: A ^ B = A;
  let a be set;
  assume a in A;
  then consider b,c be set such that
  b in A and
A2: c in B and
A3: a = b \/ c by A1,SUBSTLAT:15;
  take c;
  thus thesis by A2,A3,XBOOLE_1:7;
end;
