reserve B,C,D for Category;

theorem Th14:
  for a,b,c being Object of C st Hom(a,b) <> {} & Hom(b,c) <> {}
  for f being Morphism of a,b, g being Morphism of b,c
   holds (g(*)f) opp = (f opp)(*)(g opp)
proof
 let a,b,c be Object of C such that
A1: Hom(a,b) <> {} and
A2: Hom(b,c) <> {};
A3: Hom(b opp,a opp) <> {} by A1,Th4;
A4: Hom(c opp,b opp) <> {} by A2,Th4;
  let f be Morphism of a,b, g be Morphism of b,c;
A5: dom g = b by A2,CAT_1:5 .=  cod f by A1,CAT_1:5;
  then
A6: g(*)f = ( the Comp of C ).(g,f) by CAT_1:16;
A7: f opp = f & g opp = g by A1,A2,Def6;
A8: dom g = b opp by A2,CAT_1:5 .= cod(g opp) by A4,CAT_1:5;
A9: cod f = b opp by A1,CAT_1:5 .= dom(f opp) by A3,CAT_1:5;
  then
  the Comp of C = ~(the Comp of C opp) & [f opp,g opp] in dom(the Comp of
  C opp) by A5,A8,CAT_1:15,FUNCT_4:53;
  then (the Comp of C ).(g,f) = (the Comp of C opp).(f opp,g opp) by A7,
FUNCT_4:def 2;
  hence thesis by A5,A6,A8,A9,CAT_1:16;
end;
