
theorem Th38:
  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 retraction
  holds rng f = the_carrier_of b
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;
  given g being Morphism of b,a such that
A3: g is_right_inverse_of f;
A4: f*g = idm b by A3;
A5: f qua Function*g = f*g by A1,A2,Th36;
A6: f is Function of the_carrier_of a, the_carrier_of b by A1,Th34;
A7: g is Function of the_carrier_of b, the_carrier_of a by A2,Th34;
  idm b = id the_carrier_of b by Def10;
  hence thesis by A4,A5,A6,A7,FUNCT_2:18;
end;
