reserve n for Nat;
reserve K for Field;
reserve a,b,c,d,e,f,g,h,i,a1,b1,c1,d1,e1,f1,g1,h1,i1 for Element of K;
reserve M,N for Matrix of 3,K;
reserve p for FinSequence of REAL;
reserve a,b,c,d,e,f for Real;
reserve u,u1,u2 for non zero Element of TOP-REAL 3;
reserve P for Element of ProjectiveSpace TOP-REAL 3;
reserve a,b,c,d,e,f,g,h,i for Element of F_Real;
reserve M for Matrix of 3,F_Real;
reserve e1,e2,e3,f1,f2,f3 for Element of F_Real;
reserve MABC,MAEF,MDBF,MDEC,MDEF,MDBC,MAEC,MABF,
        MABE,MACF,MBDF,MCDE,MACE,MBDE,MCDF for Matrix of 3,F_Real;
reserve r1,r2 for Real;
reserve p1,p2,p3,p4,p5,p6 for Point of TOP-REAL 3;
reserve p7,p8,p9 for Point of TOP-REAL 3;

theorem Th23:
  |{p2,p4,p9}| = 0 implies
  |{p2,p4,p5}| * |{p2,p9,p7}| = - |{p2,p4,p7}| * |{p2,p5,p9}|
  proof
    assume
A1: |{p2,p4,p9}| = 0;
    |{p2,p4,p5}| * |{p2,p9,p7}| - |{p2,p4,p9}| * |{p2,p5,p7}| +
      |{p2,p4,p7}| * |{p2,p5,p9}| = 0 by ANPROJ_8:28;
    hence thesis by A1;
  end;
