
theorem Th41:
  for C being para-functional semi-functional category
  for a,b being Object of C st <^a,b^> <> {} for f being Morphism of a,b
  st f is one-to-one & (f qua Function)" in <^b,a^> holds f is iso
proof
  let C be para-functional semi-functional category;
  let a,b be Object of C such that
A1: <^a,b^> <> {};
  let f be Morphism of a,b;
  assume
A2: f is one-to-one;
  assume
A3: f qua Function" in <^b,a^>;
  then reconsider g = f qua Function" as Morphism of b,a;
  dom g = the_carrier_of b by A3,Th35;
  then
A4: rng f = the_carrier_of b by A2,FUNCT_1:33;
A5: f qua Function"*f = id dom f by A2,FUNCT_1:39;
A6: f*(f qua Function") = id rng f by A2,FUNCT_1:39;
A7: dom f = the_carrier_of a by A1,Th35;
A8: f*g = id the_carrier_of b by A1,A3,A4,A6,Th36;
A9: g*f = id the_carrier_of a by A1,A3,A5,A7,Th36;
A10: idm b = f*g by A8,Th37;
 idm a = g*f by A9,Th37;
  then
A11: g is_left_inverse_of f;
A12: g is_right_inverse_of f by A10;
  then f is retraction coretraction by A11;
  hence f*f" = idm b & f"*f = idm a by A1,A3,A11,A12,ALTCAT_3:def 4;
end;
