theorem
  X c= Y & Y c= Z & X,Z are_equipotent implies X,Y are_equipotent & Y,Z
  are_equipotent
proof
  assume that
A1: X c= Y & Y c= Z and
A2: X,Z are_equipotent;
A3: card X = card Z by A2,Th4;
  card X c= card Y & card Y c= card Z by A1,Th10;
  hence thesis by A3,Th4,XBOOLE_0:def 10;
end;
