
theorem
  for C being Categorial Category, f,g being Morphism of C st dom g =
  cod f holds g(*)f = [[dom f, cod g], g`2*f`2]
proof
  let C be Categorial Category;
  let f,g be Morphism of C;
A1: g`11 = dom g by CAT_5:13;
A2: f`11 = dom f by CAT_5:13;
  assume
A3: dom g = cod f;
  then consider ff being Functor of f`11, g`11 such that
A4: f = [[dom f, cod f], ff] by A1,A2,CAT_5:def 6;
A5: g`12 = cod g by CAT_5:13;
  then consider gg being Functor of g`11, g`12 such that
A6: g = [[dom g, cod g], gg] by A1,CAT_5:def 6;
  thus g(*)f = [[dom f, cod g], gg*ff] by A3,A1,A5,A2,A6,A4,CAT_5:def 6
    .= [[dom f, cod g], gg*f`2] by A4
    .= [[dom f, cod g], g`2*f`2] by A6;
end;
