
theorem Th9:
  for C being associative transitive non empty AltCatStr, o1,o2,
o3 being Object of C st <^o1,o2^> <> {} & <^o2,o3^> <> {} for A being Morphism
  of o1,o2, B being Morphism of o2,o3 st A is mono & B is mono holds B * A is
  mono
proof
  let C be associative transitive non empty AltCatStr, o1,o2,o3 be Object of
  C;
  assume that
A1: <^o1,o2^> <> {} and
A2: <^o2,o3^> <> {};
  let A be Morphism of o1,o2, B be Morphism of o2,o3;
  assume that
A3: A is mono and
A4: B is mono;
  let o be Object of C;
  assume
A5: <^o,o1^> <> {};
  then
A6: <^o,o2^> <> {} by A1,ALTCAT_1:def 2;
  let M1,M2 be Morphism of o,o1;
  assume
A7: (B*A)*M1 = (B*A)*M2;
  (B*A)*M1 = B*(A*M1) & (B*A)*M2 = B*(A*M2) by A1,A2,A5,ALTCAT_1:21;
  then A*M1 = A*M2 by A4,A7,A6;
  hence thesis by A3,A5;
end;
