reserve A,B,C,O for Ordinal,
        X for set,
        o for object,
        x,y,z,t,r,l for Surreal;

theorem Th29:
  for X1,X2,Y1,Y2 be set st
    X1 <==> X2 & Y1 <==> Y2 & x = [X1,Y1] & y = [X2,Y2]
  holds x == y
proof
  let X1,X2,Y1,Y2 be set such that
  A1:X1 <==> X2 & Y1 <==> Y2 & x = [X1,Y1] & y = [X2,Y2];
  A2: now let x;
    assume x in X1;
    then ex y1,y2 be Surreal st y1 in X2&y2 in X2 & y1 <= x <= y2 by A1,Def3;
    hence ex y be Surreal st y in X2 & x <= y;
  end;
  A3: now let x;
    assume x in Y2;
    then ex y1,y2 be Surreal st y1 in Y1 & y2 in Y1&y1 <= x <= y2 by A1,Def3;
    hence ex y be Surreal st y in Y1 & y <= x;
  end;
  A4: now let x;
    assume x in X2;
    then ex y1,y2 be Surreal st y1 in X1 &y2 in X1 & y1 <= x <= y2 by A1,Def3;
    hence ex y be Surreal st y in X1 & x <= y;
  end;
  now let x;
    assume x in Y1;
    then ex y1,y2 be Surreal st y1 in Y2 & y2 in Y2&y1 <= x <= y2 by A1,Def3;
    hence ex y be Surreal st y in Y2 & y <= x;
  end;
  hence thesis by SURREAL0:44,A1,A2,A3,A4;
end;
