
theorem
  for C being Category, a,b,c,d being Object of C, f,g,h being Morphism
of C st f in Hom(a,b) & g in Hom(b,c) & h in Hom(c,d) holds Den(compsym(a,c,d),
  MSAlg C).<*h, Den(compsym(a,b,c), MSAlg C).<*g,f*>*> = Den(compsym(a,b,d),
  MSAlg C).<*Den(compsym(b,c,d), MSAlg C).<*h,g*>, f*>
proof
  let C be Category, a,b,c,d be Object of C, f,g,h be Morphism of C;
  assume that
A1: f in Hom(a,b) and
A2: g in Hom(b,c) and
A3: h in Hom(c,d);
A4: cod g = c by A2,CAT_1:1;
A5: Den(compsym(b,c,d), MSAlg C).<*h,g*> = h(*)g by A2,A3,Th31;
A6: cod f = b by A1,CAT_1:1;
A7: dom h = c by A3,CAT_1:1;
  cod h = d by A3,CAT_1:1;
  then
A8: cod (h(*)g) = d by A4,A7,CAT_1:17;
A9: dom g = b by A2,CAT_1:1;
  then dom (h(*)g) = b by A4,A7,CAT_1:17;
  then
A10: h(*)g in Hom(b,d) by A8;
  dom f = a by A1,CAT_1:1;
  then
A11: dom (g(*)f) = a by A6,A9,CAT_1:17;
  cod (g(*)f) = c by A6,A9,A4,CAT_1:17;
  then
A12: g(*)f in Hom(a,c) by A11;
  Den(compsym(a,b,c), MSAlg C).<*g,f*> = g(*)f by A1,A2,Th31;
  hence Den(compsym(a,c,d), MSAlg C).<*h, Den(compsym(a,b,c), MSAlg C).<*g,f*>
  *> = h(*)(g(*)f) by A3,A12,Th31
    .= (h(*)g)(*)f by A6,A9,A4,A7,CAT_1:18
    .= Den(compsym(a,b,d),MSAlg C).<*Den(compsym(b,c,d),MSAlg C).<*h,g*>,f*>
  by A1,A5,A10,Th31;
end;
