
theorem v1:
for R being preordered Ring,
    P being Preordering of R holds (-P) + (-P) c= -P & (-P) * (-P) c= P
proof
let R be preordered Ring, P be Preordering of R;
X: P + P c= P & P * P c= P by REALALG1:def 14;
hereby let o be object;
  assume o in (-P) + (-P);
  then consider a,b being Element of R such that
  A: o = a + b & a in -P & b in -P;
  -a in -- P & -b in --P by A;
  then (-a) + (-b) in P + P;
  then -a + -b in P by X;
  then -(a+b) in P by RLVECT_1:31;
  then --(a+b) in -P;
  hence o in -P by A;
  end;
let o be object;
  assume o in (-P) * (-P);
  then consider a,b being Element of R such that
  A: o = a * b & a in -P & b in -P;
  -a in -- P & -b in --P by A;
  then (-a) * (-b) in P * P;
  then (-a) * (-b) in P by X;
  hence o in P by A,VECTSP_1:10;
end;
