reserve x,A,B,X,X9,Y,Y9,Z,V for set;

theorem
  X c< Y & Y c= Z implies X c< Z
proof
  assume that
A1: X c< Y and
A2: Y c= Z;
  X c= Y by A1;
  hence X c= Z & X <> Z by A1,A2,XBOOLE_0:def 10;
end;
