theorem
  x in X /\ Y implies In(x,X) = In(x,Y)
proof
  assume
A1: x in X /\ Y;
  then
A2: x in Y by XBOOLE_0:def 4;
  x in X by A1,XBOOLE_0:def 4;
  hence In (x,X) = x by Def7
    .= In (x,Y) by A2,Def7;
end;
