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 Th12:
  not X,bool X are_equipotent
proof
  given f such that
  f is one-to-one and
A1: dom f = X & rng f = bool X;
  defpred P[object] means for Y st Y = f.$1 holds not $1 in Y;
  consider Z such that
A2: for a being object holds a in Z iff a in X & P[a] from XBOOLE_0:sch 1;
  Z c= X
  by A2;
  then consider a being object such that
A3: a in X and
A4: Z = f.a by A1,FUNCT_1:def 3;
  ex Y st Y = f.a & a in Y by A2,A3,A4;
  hence contradiction by A2,A4;
end;
