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

theorem
  a, a[x][1]C are_isomorphic & a,[1]C[x]a are_isomorphic
proof
  thus a,a[x][1]C are_isomorphic
  proof
    thus Hom(a,a[x][1]C) <> {} & Hom(a[x][1]C,a) <> {} by Th19,Th28;
    take rho'(a), rho(a);
    thus thesis by Th29;
  end;
  thus Hom(a,[1]C[x]a) <> {} & Hom([1]C[x]a,a) <> {} by Th19,Th28;
  take lambda'(a), lambda(a);
  thus thesis by Th29;
end;
