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

theorem Th52:
  for o1, o2 being Object of AllIso C for m being Morphism of o1,
  o2 st <^o1,o2^> <> {} holds m is iso & m" in <^o2,o1^>
proof
  let o1, o2 be Object of AllIso C, m be Morphism of o1, o2 such that
A1: <^o1,o2^> <> {};
  reconsider p1 = o1, p2 = o2 as Object of C by Def5;
  reconsider p = m as Morphism of p1, p2 by A1,ALTCAT_2:33;
  p in (the Arrows of AllIso C).(o1,o2) by A1;
  then
A2: <^p1,p2^> <> {} & <^p2,p1^> <> {} by Def5;
A3: p is iso by A1,Def5;
  then
A4: p" is iso by A2,Th3;
  then
A5: p" in (the Arrows of AllIso C).(p2,p1) by A2,Def5;
  reconsider m1 = p" as Morphism of o2, o1 by A2,A4,Def5;
A6: m is retraction
  proof
    take m1;
    thus m * m1 = p * p" by A1,A5,ALTCAT_2:32
      .= idm p2 by A3
      .= idm o2 by ALTCAT_2:34;
  end;
A7: m is coretraction
  proof
    take m1;
    thus m1 * m = p" * p by A1,A5,ALTCAT_2:32
      .= idm p1 by A3
      .= idm o1 by ALTCAT_2:34;
  end;
  p" in <^o2,o1^> by A2,A4,Def5;
  hence m is iso by A1,A6,A7,ALTCAT_3:6;
  thus thesis by A5;
end;
