
theorem Th26:
  for C being category, o1,o2 being Object of C st o1 is initial &
  o2 is initial holds o1,o2 are_iso
proof
  let C be category, o1,o2 be Object of C such that
A1: o1 is initial and
A2: o2 is initial;
  ex N being Morphism of o2,o2 st N in <^o2,o2^> & <^o2,o2^> is trivial by A2;
  then consider y being object such that
A3: <^o2,o2^> = {y} by ZFMISC_1:131;
  consider M2 being Morphism of o2,o1 such that
A4: M2 in <^o2,o1^> and
  <^o2,o1^> is trivial by A2;
  consider M1 being Morphism of o1,o2 such that
A5: M1 in <^o1,o2^> and
  <^o1,o2^> is trivial by A1;
  thus <^o1,o2^> <> {} & <^o2,o1^> <> {} by A5,A4;
  M1 * M2 = y & idm o2 = y by A3,TARSKI:def 1;
  then M2 is_right_inverse_of M1;
  then
A6: M1 is retraction;
  ex M being Morphism of o1,o1 st M in <^o1,o1^> & <^o1,o1^> is trivial by A1;
  then consider x being object such that
A7: <^o1,o1^> = {x} by ZFMISC_1:131;
  M2 * M1 = x & idm o1 = x by A7,TARSKI:def 1;
  then M2 is_left_inverse_of M1;
  then M1 is coretraction;
  then M1 is iso by A5,A4,A6,Th6;
  hence thesis;
end;
