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 Th26:
  |{p3,p5,p7}| = 0 implies
  |{p1,p3,p5}| * |{p5,p8,p7}| = - |{p1,p5,p7}| * |{p3,p5,p8}|
  proof
    assume
A1: |{p3,p5,p7}| = 0;
A2: |{p3,p5,p7}| = - |{p5,p3,p7}| & |{p5,p8,p3}| = |{p3,p5,p8}| &
      |{p5,p7,p1}| = |{p1,p5,p7}| & |{p1,p3,p5}| = |{p5,p1,p3}|
      by ANPROJ_8:30,Th01; then
A3: |{p5,p3,p7}| = 0 & - |{p1,p3,p5}| = |{p5,p3,p1}| by ANPROJ_8:29,A1;
    |{p5,p8,p3}| * |{p5,p7,p1}| - |{p5,p8,p7}| * |{p5,p3,p1}| +
      |{p5,p8,p1}| * |{p5,p3,p7}| = 0 by ANPROJ_8:28;
    then |{p3,p5,p8}| * |{p1,p5,p7}| = |{p5,p8,p7}| * ( - |{p1,p3,p5}|)
      by A2,A3;
    hence thesis;
  end;
