 reserve a for non empty set;
 reserve b, x, o for object;

theorem Th1:
    for a holds
    ex b be object st (for x be set holds not [x,b] in a)
    proof
      let a;
reconsider b={u where u is Element of a:ex y,z be set st [y,z]=u & not u in z}
           as set;
A1:   for x be set holds [x,b] in b implies [x,b] in a & not [x,b] in b
      proof
        let x be set;
        assume [x,b] in b; then
        consider u1 be Element of a such that
A3:     u1 = [x,b] & ex y,z be set st
        [y,z] = u1 & not u1 in z;
        thus thesis by A3,XTUPLE_0:1;
      end;
A4:   for x be set holds not [x,b] in a
      proof
        let x be set;
        assume
A5:     [x,b] in a;
        per cases;
          suppose [x,b] in b;
            hence contradiction by A1;
          end;
          suppose
            not [x,b] in b;
            hence thesis by A5;
          end;
        end;
        take b;
        thus thesis by A4;
      end;
