reserve A,B for Ordinal,
  K,M,N for Cardinal,
  x,x1,x2,y,y1,y2,z,u for object,X,Y,Z,X1,X2, Y1,Y2 for set,
  f,g for Function;

theorem Th6:
  [:X,Y:],[:card X,Y:] are_equipotent & [:X,Y:],[:X,card Y:]
are_equipotent & [:X,Y:],[:card X,card Y:] are_equipotent & card [:X,Y:] = card
[:card X,Y:] & card [:X,Y:] = card [:X,card Y:] & card [:X,Y:] = card [:card X,
  card Y:]
proof
  [:Y,X:],[:card Y,X:] are_equipotent & [:X,Y:],[:Y,X:] are_equipotent by Lm2
,Lm3;
  then
A1: [:X,Y:],[:card Y,X:] are_equipotent by WELLORD2:15;
A2: [:card Y,X:],[:X,card Y:] are_equipotent by Lm2;
  hence
A3: [:X,Y:],[:card X,Y:] are_equipotent & [:X,Y:],[:X,card Y:]
  are_equipotent by A1,Lm3,WELLORD2:15;
  [:X,card Y:],[:card X,card Y:] are_equipotent by Lm3;
  hence [:X,Y:],[:card X,card Y:] are_equipotent by A3,WELLORD2:15;
  hence thesis by A2,A1,Lm3,CARD_1:5,WELLORD2:15;
end;
