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

theorem
  C is with_finite_product
proof
A1: for a,b ex c being Object of C, p1,p2 being Morphism of C st dom p1 = c
  & dom p2 = c & cod p1 = a & cod p2 = b & c is_a_product_wrt p1,p2
  proof
    let a,b;
    take a[x]b, pr1(a,b), pr2(a,b);
    thus thesis by Def8,Th17,Th18;
  end;
  [1]C is terminal by Def8;
  hence thesis by A1,Th1;
end;
