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
  SliceFunctor (g(*)f) = (SliceFunctor g)*(SliceFunctor f)
proof
  let C be Category, f,g be Morphism of C;
  assume
A1: dom g = cod f;
  then
A2: dom (g(*)f) = dom f by CAT_1:17;
  set A1 = C-SliceCat dom f, A3 = C-SliceCat cod g;
  set F = SliceFunctor f;
  reconsider G = SliceFunctor g as Functor of C-SliceCat cod f,A3 by A1;
  reconsider GF = SliceFunctor (g(*)f) as Functor of A1,A3 by A1,A2,CAT_1:17;
  now
    let m be Morphism of A1;
A3: F.m = [[f(*)m`11, f(*)m`12], m`2] by Def13;
    then
A4: (F.m)`11 = f(*)m`11 by MCART_1:85;
A5: (F.m)`12 = f(*)m`12 by A3,MCART_1:85;
A6: (F.m)`2 = m`2 by A3;
A7: dom f = cod m`11 by Th23;
A8: dom f = cod m`12 by Th23;
A9: g(*)(f(*)m`11) = g(*)f(*)m`11 by A1,A7,CAT_1:18;
A10: g(*)(f(*)m`12) = g(*)f(*)m`12 by A1,A8,CAT_1:18;
    thus (G*F).m = G.(F.m) by FUNCT_2:15
      .= [[g(*)(f(*)m`11), g(*)(f(*)m`12)], m`2] by A1,A4,A5,A6,Def13
      .= GF.m by A2,A9,A10,Def13;
  end;
  hence thesis by FUNCT_2:63;
end;
