reserve C for category,
  o1, o2, o3 for Object of C;

theorem
  (for o1, o2 being Object of C, m being Morphism of o1, o2 holds m is
  coretraction & <^o2,o1^> <> {}) implies the AltCatStr of C = AllCoretr C
proof
  assume
A1: for o1, o2 being Object of C, m being Morphism of o1, o2 holds m is
  coretraction & <^o2,o1^> <> {};
A2: the carrier of AllCoretr C = the carrier of the AltCatStr of C by Def4;
A3: the Arrows of AllCoretr C cc= the Arrows of C by Def4;
  now
    let i be object;
    assume
A4: i in [:the carrier of C,the carrier of C:];
    then consider o1, o2 being object such that
A5: o1 in the carrier of C & o2 in the carrier of C and
A6: i = [o1,o2] by ZFMISC_1:84;
    reconsider o1, o2 as Object of C by A5;
    thus (the Arrows of AllCoretr C).i = (the Arrows of C).i
    proof
      thus (the Arrows of AllCoretr C).i c= (the Arrows of C).i by A2,A3,A4;
      let n be object;
      assume
A7:   n in (the Arrows of C).i;
      then reconsider n1 = n as Morphism of o1, o2 by A6;
      <^o2,o1^> <> {} & n1 is coretraction by A1;
      then n in (the Arrows of AllCoretr C).(o1,o2) by A6,A7,Def4;
      hence thesis by A6;
    end;
  end;
  hence thesis by A2,ALTCAT_2:25,PBOOLE:3;
end;
