reserve X,Y,Z,X1,X2,Y1,Y2 for set, x,y,z,t,x1,x2 for object,
  f,g,h,f1,f2,g1,g2 for Function;

theorem Th33:
  rng f c= PFuncs(X,Y) implies rng uncurry f c= Y & rng uncurry' f c= Y
proof
  assume
A1: rng f c= PFuncs(X,Y);
  thus
A2: rng uncurry f c= Y
  proof
    let x be object;
    assume x in rng uncurry f;
    then consider y being object such that
A3: y in dom uncurry f and
A4: x = (uncurry f).y by FUNCT_1:def 3;
    consider z,g,t such that
A5: y = [z,t] and
A6: z in dom f & g = f.z and
A7: t in dom g by A3,Def2;
    g in rng f by A6,FUNCT_1:def 3;
    then
A8: ex g1 st g = g1 & dom g1 c= X & rng g1 c= Y by A1,PARTFUN1:def 3;
    (uncurry f).(z,t) = g.t & g.t in rng g by A6,A7,Th31,FUNCT_1:def 3;
    hence thesis by A4,A5,A8;
  end;
  rng uncurry' f c= rng uncurry f by FUNCT_4:41;
  hence thesis by A2;
end;
