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;

theorem Th55:
  z in X \ {x} iff z in X & z <> x
proof
  z in X \ {x} iff z in X & not z in {x} by XBOOLE_0:def 5;
  hence thesis by TARSKI:def 1;
end;
