reserve u,v,x,x1,x2,y,y1,y2,z,p,a for object,
        A,B,X,X1,X2,X3,X4,Y,Y1,Y2,Z,N,M for set;
reserve e for object, X,X1,X2,Y1,Y2 for set;

theorem
 [x,y] in X implies x in union union X & y in union union X
proof
  assume
A1: [x,y] in X;
  { x } in {{x},{x,y}} by TARSKI:def 2;
  then
A2: {x} in union X by A1,TARSKI:def 4;
  {x,y} in {{x},{x,y}} by TARSKI:def 2;
  then
A3: {x,y} in union X by A1,TARSKI:def 4;
  y in {x,y} & x in {x} by TARSKI:def 1,def 2;
  hence thesis by A3,A2,TARSKI:def 4;
end;
