reserve x,x1,x2,x3,x4,y,y1,y2,y3,y4,z,z1,z2,z2,z4 for object;

theorem Th1:
  [x1,x2] = [y1,y2] implies x1 = y1 & x2 = y2
proof
  assume
A1: [x1,x2] = [y1,y2];
  per cases;
  suppose
A2: y1 <> y2;
    then
A3: {x1} <> {y1,y2} by Lm2;
    then {x1} = {y1} by A1,Lm3;
   then x1 in {y1} by TARSKI:def 1;
   hence
A4: x1 = y1 by TARSKI:def 1;
    {y1,y2} = {x1,x2} by A1,A3,Lm3;
   hence thesis by A2,A4,Lm3;
  end;
  suppose
A5: y1 = y2;
    then {{x1,x2},{x1}} = {{y1},{y1}} by A1,ENUMSET1:29
        .= {{y1}} by ENUMSET1:29;
    then { y1 } = { x1,x2 } by Lm1;
   hence thesis by A5,Lm1;
  end;
end;
