theorem
  X,Y are_equipotent or card X = card Y implies bool X,bool Y
  are_equipotent & card bool X = card bool Y
proof
  assume X,Y are_equipotent or card X = card Y;
  then X,Y are_equipotent by CARD_1:5;
  then
A1: card Funcs(X,{0,1}) = card Funcs(Y,{0,1}) by FUNCT_5:60;
  card Funcs(X,{0,1}) = card bool X & card Funcs(Y,{0,1}) = card bool Y by
FUNCT_5:65;
  hence thesis by A1,CARD_1:5;
end;
