reserve A,B,C for Ordinal,
  K,L,M,N for Cardinal,
  x,y,y1,y2,z,u for object,X,Y,Z,Z1,Z2 for set,
  n for Nat,
  f,f1,g,h for Function,
  Q,R for Relation;
reserve ff for Cardinal-Function;

theorem Th11:
  Funcs(X,Y) = product (X --> Y)
proof
  set f = (X --> Y);
  thus Funcs(X,Y) c= product f
  proof
    let x be object;
    assume x in Funcs(X,Y);
    then consider g such that
A2: x = g and
A3: dom g = X and
A4: rng g c= Y by FUNCT_2:def 2;
    now
      let y be object;
      assume
A5:   y in dom f;
      then
A6:   f.y = Y by FUNCOP_1:7;
      g.y in rng g by A3,A5,FUNCT_1:def 3;
      hence g.y in f.y by A4,A6;
    end;
    hence thesis by A2,A3,Def5;
  end;
  let x be object;
  assume x in product f;
  then consider g such that
A7: x = g and
A8: dom g = dom f and
A9: for x being object st x in dom f holds g.x in f.x by Def5;
  rng g c= Y
  proof
    let y be object;
    assume y in rng g;
    then consider z being object such that
A10: z in dom g and
A11: y = g.z by FUNCT_1:def 3;
    y in f.z by A8,A9,A10,A11;
    hence thesis by A8,A10,FUNCOP_1:7;
  end;
  hence thesis by A7,A8,FUNCT_2:def 2;
end;
