reserve e for set;
reserve C,C1,C2,C3 for AltCatStr;
reserve C for non empty AltCatStr,
  o for Object of C;
reserve C for non empty transitive AltCatStr;

theorem Th32:
  for C being non empty transitive AltCatStr, D being non empty
transitive SubCatStr of C, p1,p2,p3 being Object of D st <^p1,p2^> <> {} & <^p2
,p3^> <> {} for o1,o2,o3 being Object of C st o1 = p1 & o2 = p2 & o3 = p3 for f
being Morphism of o1,o2, g being Morphism of o2,o3, ff being Morphism of p1,p2,
  gg being Morphism of p2,p3 st f = ff & g = gg holds g*f = gg*ff
proof
  let C be non empty transitive AltCatStr, D be non empty transitive SubCatStr
  of C;
  let p1,p2,p3 be Object of D such that
A1: <^p1,p2^> <> {} & <^p2,p3^> <> {};
  let o1,o2,o3 be Object of C such that
A2: o1 = p1 & o2 = p2 & o3 = p3;
  let f be Morphism of o1,o2, g be Morphism of o2,o3, ff be Morphism of p1,p2,
  gg be Morphism of p2,p3 such that
A3: f = ff & g = gg;
  <^p1,p3^> <> {} by A1,ALTCAT_1:def 2;
  then dom((the Comp of D).(p1,p2,p3)) = [:<^p2,p3^>,<^p1,p2^>:] by
FUNCT_2:def 1;
  then
A4: [gg,ff] in dom((the Comp of D).(p1,p2,p3)) by A1,ZFMISC_1:87;
A5: the Comp of D cc= the Comp of C by Def11;
  (the Comp of D).(p1,p2,p3) = (the Comp of D).[p1,p2,p3] & (the Comp of C
  ).( o1,o2,o3) = (the Comp of C).[o1,o2,o3] by MULTOP_1:def 1;
  then
A6: (the Comp of D).(p1,p2,p3) c= (the Comp of C).(o1,o2,o3) by A2,A5;
  <^o1,o2^> <> {} & <^o2,o3^> <> {} by A1,A2,Th31,XBOOLE_1:3;
  hence g*f = (the Comp of C).(o1,o2,o3).(g,f) by ALTCAT_1:def 8
    .= (the Comp of D).(p1,p2,p3).(gg,ff) by A3,A4,A6,GRFUNC_1:2
    .= gg*ff by A1,ALTCAT_1:def 8;
end;
