
theorem abs10:
for R being preordered non degenerated Ring,
    P being Preordering of R,
    a,b being P-ordered Element of R holds abs(P,a*b) = abs(P,a) * abs(P,b)
proof
let R be preordered non degenerated Ring, P be Preordering of R,
    a,b be P-ordered Element of R;
  AS: (a in P \/ -P & b in P \/ -P) by defppp;
  X: P * P c= P by REALALG1:def 14;
  Y: (-P) * P c= -P & P * (-P) c= -P by v2;
  Z: (-P) * (-P) c= P by v1;
  per cases by AS,XBOOLE_0:def 3;
  suppose A: a in P;
    per cases by AS,XBOOLE_0:def 3;
    suppose A1: b in P;
      then B: abs(P,a) = a & abs(P,b) = b by A,defa;
      a * b in P * P by A,A1;
      hence abs(P,a) * abs(P,b) = abs(P,a*b) by X,defa,B;
      end;
    suppose A1: b in -P;
      then B: abs(P,a) = a & abs(P,b) = -b by A,defa;
      C: a * b in P * (-P) by A,A1;
      thus abs(P,a) * abs(P,b) = -(a*b) by VECTSP_1:8,B
                              .= abs(P,a*b) by C,Y,defa;
      end;
    end;
  suppose A: a in -P;
    per cases by AS,XBOOLE_0:def 3;
    suppose A1: b in -P;
      then B: abs(P,a) = -a & abs(P,b) = -b by A,defa;
      C: a * b in (-P) * (-P) by A,A1;
      thus abs(P,a) * abs(P,b) = a * b by VECTSP_1:10,B
                              .= abs(P,a*b) by C,Z,defa;
      end;
    suppose A1: b in P;
      then B: abs(P,a) = -a & abs(P,b) = b by A,defa;
      C: a * b in (-P) * P by A,A1;
      thus abs(P,a) * abs(P,b) = -(a*b) by VECTSP_1:9,B
                              .= abs(P,a*b) by C,Y,defa;
      end;
    end;
end;
