
theorem Th12:
  for X,Y,Z being non empty TopSpace for f being continuous
  Function of Y,Z for x being Element of X, A being Subset of oContMaps(X, Y)
  holds pi(oContMaps(X,f).:A, x) = f.:pi(A, x)
proof
  let X,Y,Z be non empty TopSpace;
  let f be continuous Function of Y,Z;
  set Xf = oContMaps(X,f);
  let x be Element of X, A be Subset of oContMaps(X, Y);
  thus pi(Xf.:A,x) c= f.:pi(A,x)
  proof
    let a be object;
    assume a in pi(Xf.:A,x);
    then consider h being Function such that
A1: h in Xf.:A and
A2: a = h.x by CARD_3:def 6;
    consider g being object such that
A3: g in the carrier of oContMaps(X,Y) and
A4: g in A and
A5: h = Xf.g by A1,FUNCT_2:64;
    reconsider g as continuous Function of X,Y by A3,Th2;
    h = f*g by A5,Def2;
    then
A6: a = f.(g.x) by A2,FUNCT_2:15;
    g.x in pi(A,x) by A4,CARD_3:def 6;
    hence thesis by A6,FUNCT_2:35;
  end;
  let a be object;
  assume a in f.:pi(A,x);
  then consider b being object such that
  b in the carrier of Y and
A7: b in pi(A,x) and
A8: a = f.b by FUNCT_2:64;
  consider g being Function such that
A9: g in A and
A10: b = g.x by A7,CARD_3:def 6;
  reconsider g as continuous Function of X,Y by A9,Th2;
  f*g = Xf.g by Def2;
  then
A11: f*g in Xf.:A by A9,FUNCT_2:35;
  a = (f*g).x by A8,A10,FUNCT_2:15;
  hence thesis by A11,CARD_3:def 6;
end;
