
theorem
  for C being category, o1, o2, o3, o4 being Object of C for a being
Morphism of o1, o2, b being Morphism of o2, o3 for c being Morphism of o1, o4,
d being Morphism of o4, o3 st b*a = d*c & a*(a") = idm o2 & d"*d = idm o4 & <^
o1,o2^> <> {} & <^o2,o1^> <> {} & <^o2,o3^> <> {} & <^o3,o4^> <> {} & <^o4,o3^>
  <> {} holds c*(a") = d"*b
proof
  let C be category, o1, o2, o3, o4 be Object of C, a be Morphism of o1, o2, b
  be Morphism of o2, o3, c be Morphism of o1, o4, d be Morphism of o4, o3 such
  that
A1: b*a = d*c and
A2: a*(a") = idm o2 and
A3: d"*d = idm o4 and
A4: <^o1,o2^> <> {} and
A5: <^o2,o1^> <> {} and
A6: <^o2,o3^> <> {} and
A7: <^o3,o4^> <> {} and
A8: <^o4,o3^> <> {};
A9: <^o2,o4^> <> {} by A6,A7,ALTCAT_1:def 2;
  <^o1,o3^> <> {} by A4,A6,ALTCAT_1:def 2;
  then
A10: <^o1,o4^> <> {} by A7,ALTCAT_1:def 2;
  b = b*idm o2 by A6,ALTCAT_1:def 17
    .= b*a*(a") by A2,A4,A5,A6,ALTCAT_1:21;
  hence d"*b = d"*(d*(c*(a"))) by A1,A5,A8,A10,ALTCAT_1:21
    .= (d"*d)*(c*(a")) by A7,A8,A9,ALTCAT_1:21
    .= c*(a") by A3,A9,ALTCAT_1:20;
end;
