theorem Th31:
  ||.-x.|| = ||.x.||
proof
A1: |.-1.| = -(-1) by ABSVALUE:def 1;
  ||.-x.|| = ||.(-1) * x.|| by RLVECT_1:16
    .= 1 * ||.x.|| by A1,Th27;
  hence thesis;
end;
