
theorem Th39:
  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 coretraction holds f is one-to-one
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_left_inverse_of f;
A4: g*f = idm a by A3;
A5: g qua Function*f = g*f by A1,A2,Th36;
A6: dom f = the_carrier_of a by A1,Th35;
  idm a = id the_carrier_of a by Def10;
  hence thesis by A4,A5,A6,FUNCT_1:31;
end;
