reserve I for set;

theorem Th3: :: see PRALG_2:5
  for A, B being non empty set, C, y being set for f being Function
  st f in Funcs(A,Funcs(B,C)) & y in B holds dom ((commute f).y) = A & rng ((
  commute f).y) c= C
proof
  let A, B be non empty set, C, y be set, f be Function such that
A1: f in Funcs(A,Funcs(B,C)) and
A2: y in B;
  set cf = commute f;
  cf in Funcs(B,Funcs(A,C)) by A1,FUNCT_6:55;
  then
A3: ex g being Function st g = cf & dom g = B & rng g c= Funcs(A,C) by
FUNCT_2:def 2;
  then cf.y in rng cf by A2,FUNCT_1:def 3;
  then ex t being Function st t = ((commute f).y) & dom t = A & rng t c= C by
A3,FUNCT_2:def 2;
  hence thesis;
end;
