theorem Th30:
  |{p2,p8,p7}| * |{p5,p9,p7}| = |{p2,p9,p7}| * |{p5,p8,p7}| implies 
  |{p7,p2,p5}| * |{p7,p8,p9}| = 0
  proof
    assume
A1: |{p2,p8,p7}| * |{p5,p9,p7}| = |{p2,p9,p7}| * |{p5,p8,p7}|;
A2: |{p7,p2,p8}| = |{p2,p8,p7}| & |{p7,p5,p9}| = |{p5,p9,p7}| &
    |{p7,p2,p9}| = |{p2,p9,p7}| & |{p7,p5,p8}| = |{p5,p8,p7}| by Th01;
    |{p7,p5,p8}| = |{p5,p8,p7}| by Th01
                .= - |{p8,p5,p7}| by ANPROJ_8:30;
    then
A3: |{p7,p5,p8}| = - |{p7,p8,p5}| by Th01;
     |{p7,p2,p8}| * |{p7,p5,p9}| - |{p7,p2,p5}| * |{p7,p8,p9}| +
       |{p7,p2,p9}| * |{p7,p8,p5}| = 0 by ANPROJ_8:28;
    then |{p7,p2,p8}| * |{p7,p5,p9}| - |{p7,p2,p5}| * |{p7,p8,p9}| -
      |{p7,p2,p9}| * |{p7,p5,p8}| = 0 by A3;
    hence thesis by A2,A1;
  end;
