reserve C for Category,
  C1,C2 for Subcategory of C;

theorem
  for C being Category, f,g being Morphism of C st dom g = cod f holds
  SliceContraFunctor (g(*)f) = (SliceContraFunctor f)*(SliceContraFunctor g)
proof
  let C be Category, f,g be Morphism of C;
  assume
A1: dom g = cod f;
  then
A2: cod (g(*)f) = cod g by CAT_1:17;
  set A1 = (dom f)-SliceCat C, A2 = (cod f)-SliceCat C;
  set A3 = (cod g)-SliceCat C;
  reconsider F=SliceContraFunctor f as Functor of A2,A1;
  reconsider G = SliceContraFunctor g as Functor of A3,A2 by A1;
  reconsider FG = SliceContraFunctor (g(*)f) as Functor of A3,A1 by A1,A2,
CAT_1:17;
  now
    let m be Morphism of A3;
A3: G.m = [[m`11(*)g, m`12(*)g], m`2] by Def14;
    then
A4: (G.m)`11 = m`11(*)g by MCART_1:85;
A5: (G.m)`12 = m`12(*)g by A3,MCART_1:85;
A6: (G.m)`2 = m`2 by A3;
A7: cod g = dom m`11 by Th24;
A8: cod g = dom m`12 by Th24;
A9: m`11(*)(g(*)f) = m`11(*)g(*)f by A1,A7,CAT_1:18;
A10: m`12(*)(g(*)f) = m`12(*)g(*)f by A1,A8,CAT_1:18;
    thus (F*G).m = F.(G.m) by FUNCT_2:15
      .= [[m`11(*)g(*)f, m`12(*)g(*)f], m`2] by A4,A5,A6,Def14
      .= FG.m by A2,A9,A10,Def14;
  end;
  hence thesis by FUNCT_2:63;
end;
