
theorem v1a:
for R being preordered Ring,
    P being Preordering of R holds (-P) * P = P * (-P)
proof
let R be preordered Ring, P be Preordering of R;
A: now let o be object;
   assume o in (-P) * P;
   then consider a,b being Element of R such that
   A1: o = a * b & a in -P & b in P;
   consider c being Element of R such that
   A2: a = -c & c in P by A1;
   A3: -b in -P by A1;
   c * (-b) = -(c * b) by VECTSP_1:8
           .= a * b by A2,VECTSP_1:9;
   hence o in P * (-P) by A1,A2,A3;
   end;
now let o be object;
   assume o in P * (-P);
   then consider a,b being Element of R such that
   A1: o = a * b & a in P & b in -P;
   consider c being Element of R such that
   A2: b = -c & c in P by A1;
   A3: -a in -P by A1;
   (-a) * c = -(a * c) by VECTSP_1:9
           .= a * b by A2,VECTSP_1:8;
   hence o in (-P) * P by A1,A2,A3;
   end;
hence thesis by A,TARSKI:2;
end;
