
theorem
  for C being concrete category
  for a,b being Object of C st <^a,b^> <> {} & <^b,a^> <> {}
  for f being Morphism of a,b st f is iso holds f" = (f qua Function)"
proof
  let C be concrete category;
  let a,b be Object of C;
  assume that
A1: <^a,b^> <> {} and
A2: <^b,a^> <> {};
  let f be Morphism of a,b;
  assume
A3: f is iso;
  then
A4: f"*f = idm a;
A5: f"*(f qua Function) = f"*f by A1,A2,Th36;
A6: dom (f") = the_carrier_of b by A2,Th35;
A7: dom f = the_carrier_of a by A1,Th35;
A8: f is one-to-one by A1,A2,A3,Th40;
A9: rng f = the_carrier_of b by A1,A2,A3,Th40;
  idm a = id the_carrier_of a by Def10;
  hence thesis by A4,A5,A6,A7,A8,A9,FUNCT_1:41;
end;
