reserve A,B,C for Ordinal,
  X,X1,Y,Y1,Z for set,a,b,b1,b2,x,y,z for object,
  R for Relation,
  f,g,h for Function,
  k,m,n for Nat;
reserve M,N for Cardinal;

theorem Th4:
  X,Y are_equipotent iff card X = card Y
proof
A1: Y,card Y are_equipotent by Def2;
A2: X,card X are_equipotent by Def2;
  hence X,Y are_equipotent implies card X = card Y by Def2,WELLORD2:15;
  assume card X = card Y;
  hence thesis by A2,A1,WELLORD2:15;
end;
