
theorem
  for C being category, o1,o2,o3 being Object of C for M1 being Morphism
of o1,o2, M2 being Morphism of o2,o3 st M1 is _zero & M2 is _zero holds M2 * M1
  is _zero
proof
  let C be category, o1,o2,o3 be Object of C, M1 be Morphism of o1,o2, M2 be
  Morphism of o2,o3;
  assume that
A1: M1 is _zero and
A2: M2 is _zero;
  let o be Object of C;
  assume
A3: o is _zero;
  then
A4: o is initial;
  then consider B1 being Morphism of o,o2 such that
A5: B1 in <^o,o2^> and
  <^o,o2^> is trivial;
  let A be Morphism of o1,o, B be Morphism of o,o3;
  consider B2 being Morphism of o,o3 such that
A6: B2 in <^o,o3^> and
A7: <^o,o3^> is trivial by A4;
  consider y being object such that
A8: <^o,o3^> = {y} by A6,A7,ZFMISC_1:131;
A9: o is terminal by A3;
  then consider A1 being Morphism of o1,o such that
A10: A1 in <^o1,o^> and
A11: <^o1,o^> is trivial;
  consider x being object such that
A12: <^o1,o^> = {x} by A10,A11,ZFMISC_1:131;
  ex M being Morphism of o,o st M in <^o,o^> & <^o,o^> is trivial by A9;
  then consider z being object such that
A13: <^o,o^> = {z} by ZFMISC_1:131;
  consider A2 being Morphism of o2,o such that
A14: A2 in <^o2,o^> and
  <^o2,o^> is trivial by A9;
A15: idm o = z & A2 * B1 = z by A13,TARSKI:def 1;
A16: B = y & B2 = y by A8,TARSKI:def 1;
A17: A = x & A1 = x by A12,TARSKI:def 1;
A18: <^o2,o3^> <> {} by A6,A14,ALTCAT_1:def 2;
  M2 = B2 * A2 by A2,A3;
  hence M2 * M1 = (B2*A2) * (B1*A1) by A1,A3
    .= B2*A2 * B1*A1 by A5,A10,A18,ALTCAT_1:21
    .= B*(idm o)*A by A5,A6,A14,A17,A16,A15,ALTCAT_1:21
    .= B*A by A6,ALTCAT_1:def 17;
end;
