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;
