reserve U for Universe;
reserve x for Element of U;
reserve U1,U2 for Universe;

theorem
  for U1,U2 being Universe
  for x being Set of U1 for y being Set of U2 holds
  ex z being Set of sup(U1,U2) st
  for a being object holds a in z iff a = x or a = y
  proof
    let U1,U2 be Universe;
    let x be Set of U1;
    let y be Set of U2;
    consider z being set such that
A1: for a being object holds
    a in z iff a = x or a = y by TARSKI_0:3;
    per cases by CLASSES2:52;
    suppose
A2:   U1 in U2;
      then
A3:   sup(U1,U2) = U2 by Def13;
      z in U2
      proof
        assume
A4:     not z in U2;
        set X = {x,y};
        x is U1-set & y is U2-set by Def10;
        then x in U2 & y in U2 by A2,CLASSES4:def 1;
        then {x,y} in U2 by CLASSES2:58;
        hence thesis by A1,A4,TARSKI:def 2;
      end;
      then z is U2-set;
      then reconsider z as Set of sup(U1,U2) by A3,Def10;
      take z;
      thus thesis by A1;
    end;
    suppose
A5:   U2 in U1;
      then
A6:   sup(U1,U2) = U1 by Def13;
      z in U1
      proof
        assume
A7:     not z in U1;
        x is U1-set & y is U2-set by Def10;
        then x in U1 & y in U1 by A5,CLASSES4:def 1;
        then {x,y} in U1 by CLASSES2:58;
        hence thesis by A1,A7,TARSKI:def 2;
      end;
      then z is U1-set;
      then reconsider z as Set of sup(U1,U2) by A6,Def10;
      take z;
      thus thesis by A1;
    end;
    suppose
A8:   U1 = U2;
      then
A9:   sup(U1,U2) = U1 by Def13;
      z in U1
      proof
        assume
A10:    not z in U1;
        x is U1-set & y is U2-set by Def10;
        then {x,y} in U1 by A8,CLASSES2:58;
        hence thesis by A1,A10,TARSKI:def 2;
      end;
      then z is U1-set;
      then reconsider z as Set of sup(U1,U2) by A9,Def10;
      take z;
      thus thesis by A1;
    end;
end;
