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 Th27:
  for r125,r136,r246,r345,r126,r135,r245,r346,r157,
  r259,r597,r368,r268,r297,r247,r287,r358,r587 being non zero Real st
  r125 * r136 * r246 * r345 = r126 * r135 * r245 * r346 &
  r157 * r259 = - r125 * r597 & r126 * r368 =   r136 * r268 &
  r245 * r297 = - r247 * r259 & r247 * r268 = - r246 * r287 &
  r346 * r358 = r345 * r368 & r135 * r587 = - r157 * r358 holds
  r287 * r597 = r297 * r587
  proof
    let r125,r136,r246,r345,r126,r135,r245,r346,r157,
    r259,r597,r368,r268,r297,r247,r287,r358,r587 being non zero Real;
    assume that
A1: r125 * r136 * r246 * r345 = r126 * r135 * r245 * r346 and
A2: r157 * r259 = - r125 * r597 and
A3: r126 * r368 =   r136 * r268 and
A4: r245 * r297 = - r247 * r259 and
A5: r247 * r268 = - r246 * r287 and
A6: r346 * r358 = r345 * r368 and
A7: r135 * r587 = - r157 * r358;
    reconsider r1 = r246 * r125 * r247 * r259 * r136 * r268 *
    r345 * r368 * r157 * r358 * r126 * r135 * r245 * r346,
    r2 = r247 * r268 * r157 * r259 * r245 * r126 * r368 * r346 *
    r358 * r135 * r125 * r136 * r246 * r345 as non zero Real;
    (((((- r287 * r246) * (- r125 * r597)) *
    (- r247 * r259) * (r136 * r268)) * (r345 * r368)) *
    (-r157 * r358)) * (r126 * r135 * r245 * r346)
      = ((((r247 * r268 * (r157 * r259)) * (r245 * r297) * (r126 * r368)) *
      (r346 * r358)) * (r135 * r587)) * (r125 * r136 * r246 * r345)
      by A1,A2,A3,A4,A5,A6,A7;
    then r1 * (r287 * r597) = r2 * (r297 * r587) & r1 <> 0 & r2 <> 0
      by ORDINAL1:def 14;
    hence thesis by XCMPLX_1:5;
  end;
