reserve H,S for ZF-formula,
  x for Variable,
  X,Y for set,
  i for Element of NAT,
  e,u for set;
reserve M,M1,M2 for non empty set,
  f for Function,
  v1 for Function of VAR,M1,
  v2 for Function of VAR,M2,
  F,F1,F2 for Subset of WFF,
  W for Universe,
  a,b,c for Ordinal of W,
  A,B,C for Ordinal,
  L for DOMAIN-Sequence of W,
  va for Function of VAR,L.a,
  phi,xi for Ordinal-Sequence of W;

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;
