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;

theorem Th20:
  MABE = <* p1, p2, p5 *> &
  MACF = <* p1, p3, p6 *> &
  MBDF = <* p2, p4, p6 *> &
  MCDE = <* p3, p4, p5 *> &
  MABF = <* p1, p2, p6 *> &
  MACE = <* p1, p3, p5 *> &
  MBDE = <* p2, p4, p5 *> &
  MCDF = <* p3, p4, p6 *> implies
  Det MABE = |{ p1, p2, p5 }| &
  Det MACF = |{ p1, p3, p6 }| &
  Det MBDF = |{ p2, p4, p6 }| &
  Det MCDE = |{ p3, p4, p5 }| &
  Det MABF = |{ p1, p2, p6 }| &
  Det MACE = |{ p1, p3, p5 }| &
  Det MBDE = |{ p2, p4, p5 }| &
  Det MCDF = |{ p3, p4, p6 }|
  proof
    assume that
A1: MABE = <* p1, p2, p5 *> and
A2: MACF = <* p1, p3, p6 *> and
A3: MBDF = <* p2, p4, p6 *> and
A4: MCDE = <* p3, p4, p5 *> and
A5: MABF = <* p1, p2, p6 *> and
A6: MACE = <* p1, p3, p5 *> and
A7: MBDE = <* p2, p4, p5 *> and
A8: MCDF = <* p3, p4, p6 *>;
    p1 = <* p1`1,p1`2,p1`3 *> & p2 = <* p2`1,p2`2,p2`3 *> &
    p3 = <* p3`1,p3`2,p3`3 *> & p4 = <* p4`1,p4`2,p4`3 *> &
    p5 = <* p5`1,p5`2,p5`3 *> & p6 = <* p6`1,p6`2,p6`3 *> by EUCLID_5:3;
    hence thesis by A1,A2,A3,A4,A5,A6,A7,A8,ANPROJ_8:35;
  end;
