reserve o,m for set;
reserve C for Cartesian_category;
reserve a,b,c,d,e,s for Object of C;
reserve C for Cocartesian_category;
reserve a,b,c,d,e,s for Object of C;

theorem
  for f being Morphism of a,c, h being Morphism of b,d, g being Morphism
  of c,e, k being Morphism of d,s st Hom(a,c) <> {} & Hom(b,d) <> {} & Hom(c,e)
  <> {} & Hom(d,s) <> {} holds (g+k)*(f+h) = (g*f)+(k*h)
proof
  let f be Morphism of a,c, h be Morphism of b,d;
  let g be Morphism of c,e, k be Morphism of d,s;
  assume that
A1: Hom(a,c) <> {} and
A2: Hom(b,d) <> {} and
A3: Hom(c,e) <> {} and
A4: Hom(d,s) <> {};
A5: Hom(s,e+s) <> {} by Th61;
  then
A6: Hom(d,e+s) <> {} by A4,CAT_1:24;
A7: Hom(e,e+s) <> {} by Th61;
  then in1(e,s)*g*f = in1(e,s)*(g*f) by A1,A3,CAT_1:25;
  then
A8: (g*f)+(k*h) = [$in1(e,s)*g*f,in2(e,s)*k*h$] by A2,A4,A5,CAT_1:25;
  Hom(c,e+s) <> {} by A3,A7,CAT_1:24;
  hence thesis by A1,A2,A6,A8,Th75;
end;
