proposition: Lemma natmul_snum_subproof (xnz nnz : nullity) (xr nr : reality) (x : {num R & xnz & xr}) (n : {compare 0%N & nnz & nr}) (rnz := mul_nonzero_subdef xnz nnz xr nr) (rrl := mul_reality_subdef xnz nnz xr nr) : Signed.spec 0 rnz rrl (x%:num *+ n%:num).
proof: Proof. rewrite {}/rnz {}/rrl; apply/andP; split. by move: xr nr xnz nnz x n => [[[]|]|] [[[]|]|] [] []// x n; rewrite mulrn_eq0//= ?eq0F. move: xr nr xnz nnz x n => [[[]|]|] [[[]|]|] [] []/= x [[|n]//= _] //; do ?[by case: (bottom x)|by case: (bottom n) |by rewrite mulrn_wge0|by rewrite mulrn_wle0|by rewrite eq0 mul0rn |by apply: real_comparable; rewrite ?real0 ?realrMn]. Qed.
imports: HB(structures) Coq(ssreflect ssrfun ssrbool) mathcomp(ssrnat eqtype choice order ssralg ssrnum ssrint) mathcomp(mathcomp_extra)