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 Th67:
  for f being Morphism of a,c, g being Morphism of b,c, h being
Morphism of c,d st Hom(a,c)<>{} & Hom(b,c)<>{} & Hom(c,d)<>{} holds [$h*f,h*g$]
  = h*[$f,g$]
proof
  let f be Morphism of a,c, g be Morphism of b,c, h be Morphism of c,d;
  assume that
A1: Hom(a,c)<>{} & Hom(b,c)<>{} and
A2: Hom(c,d)<>{};
A3: Hom(a+b,c) <> {} by A1,Th65;
A4: Hom(b,a+b) <> {} by Th61;
  h*([$f,g$]*in2(a,b)) = h*g by A1,Def28;
  then
A5: h*[$f,g$]*in2(a,b) = h*g by A2,A4,A3,CAT_1:25;
A6: Hom(a,a+b) <> {} by Th61;
A7: Hom(a,d) <> {} & Hom(b,d) <> {} by A1,A2,CAT_1:24;
  h*([$f,g$]*in1(a,b)) = h*f by A1,Def28;
  then h*[$f,g$]*in1(a,b) = h*f by A2,A6,A3,CAT_1:25;
  hence thesis by A5,A7,Def28;
end;
