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 in X implies (X\{x})\/{x}=X
proof
  assume
A1: x in X;
  thus (X\{x})\/{x}c=X
  proof
    let y be object;
    assume y in (X\{x})\/{x};
    then y in X\{x} or y in {x} by XBOOLE_0:def 3;
    hence thesis by A1,Th55,TARSKI:def 1;
  end;
  thus X c= (X\{x})\/{x}
  proof
    let y be object;
    assume y in X;
    then y in {x} or y in X\{x} by XBOOLE_0:def 5;
    hence thesis by XBOOLE_0:def 3;
  end;
end;
