
theorem Th14:
  for X,Y,Z being non empty TopSpace for f being continuous
  Function of Y,Z for x being Element of Y, A being Subset of oContMaps(Z, X)
  holds pi(oContMaps(f, X).:A, x) = pi(A, f.x)
proof
  let X,Y,Z be non empty TopSpace;
  let f be continuous Function of Y,Z;
  set fX = oContMaps(f, X);
  let x be Element of Y, A be Subset of oContMaps(Z, X);
  thus pi(fX.:A,x) c= pi(A,f.x)
  proof
    let a be object;
    assume a in pi(fX.:A,x);
    then consider h being Function such that
A1: h in fX.:A and
A2: a = h.x by CARD_3:def 6;
    consider g being object such that
A3: g in the carrier of oContMaps(Z,X) and
A4: g in A and
A5: h = fX.g by A1,FUNCT_2:64;
    reconsider g as continuous Function of Z,X by A3,Th2;
    h = g*f by A5,Def3;
    then a = g.(f.x) by A2,FUNCT_2:15;
    hence thesis by A4,CARD_3:def 6;
  end;
  let a be object;
  assume a in pi(A,f.x);
  then consider g being Function such that
A6: g in A and
A7: a = g.(f.x) by CARD_3:def 6;
  reconsider g as continuous Function of Z,X by A6,Th2;
  g*f = fX.g by Def3;
  then
A8: g*f in fX.:A by A6,FUNCT_2:35;
  a = (g*f).x by A7,FUNCT_2:15;
  hence thesis by A8,CARD_3:def 6;
end;
