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
  a+b,b+a are_isomorphic
proof
A1: Hom(a,b+a) <> {} & Hom(b,b+a) <> {} by Th61;
  hence
A2: Hom(a+b,b+a)<>{} by Th65;
A3: Hom(a,a+b) <> {} & Hom(b,a+b) <> {} by Th61;
  hence
A4: Hom(b+a,a+b)<>{} by Th65;
  take f9 = [$in2(b,a),in1(b,a)$], f = [$in2(a,b),in1(a,b)$];
  thus f9*f = [$f9*in2(a,b),f9*in1(a,b)$] by A2,A3,Th67
    .= [$in1(b,a),f9*in1(a,b)$] by A1,Def28
    .= [$in1(b,a),in2(b,a)$] by A1,Def28
    .= id(b+a) by Th66;
  thus f*f9 = [$f*in2(b,a),f*in1(b,a)$] by A1,A4,Th67
    .= [$in1(a,b),f*in1(b,a)$] by A3,Def28
    .= [$in1(a,b),in2(a,b)$] by A3,Def28
    .= id(a+b) by Th66;
end;
