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 Th28:
  p1 = <* 1,0,0 *> &
  p2 = <* 0,1,0 *> &
  p3 = <* 0,0,1 *> &
  p4 = <* 1,1,1 *> &
  p5 = <* e1,e2,e3 *> &
  p6 = <* f1,f2,f3 *> &
  qfconic(0,0,0,r1,r2,-(r1+r2),p5) = 0 &
  qfconic(0,0,0,r1,r2,-(r1+r2),p6) = 0
  implies
  qfconic(0,0,0,r1,r2,-(r1+r2),p1) = 0 &
  qfconic(0,0,0,r1,r2,-(r1+r2),p2) = 0 &
  qfconic(0,0,0,r1,r2,-(r1+r2),p3) = 0 &
  qfconic(0,0,0,r1,r2,-(r1+r2),p4) = 0 &
  (r1 * e1 * e2 + r2 * e1 * e3 = (r1 + r2) * e2 * e3 &
  r1 * f1 * f2 + r2 * f1 * f3 = (r1 + r2) * f2 * f3)
  proof
    assume that
A1: p1 = <* 1,0,0 *> and
A2: p2 = <* 0,1,0 *> and
A3: p3 = <* 0,0,1 *> and
A4: p4 = <* 1,1,1 *> and
A5: p5 = <* e1,e2,e3 *> and
A6: p6 = <* f1,f2,f3 *> and
A8: qfconic(0,0,0,r1,r2,-(r1+r2),p5) = 0 and
A9: qfconic(0,0,0,r1,r2,-(r1+r2),p6) = 0;
    p1.1 = 1 & p1.2 = 0 & p1.3 = 0 by A1,FINSEQ_1:45;
    hence qfconic(0,0,0,r1,r2,-(r1+r2),p1) = 0;
    p2.1 = 0 & p2.2 = 1 & p2.3 = 0 by A2,FINSEQ_1:45;
    hence qfconic(0,0,0,r1,r2,-(r1+r2),p2) = 0;
    p3.1 = 0 & p3.2 = 0 & p3.3 = 1 by A3,FINSEQ_1:45;
    hence qfconic(0,0,0,r1,r2,-(r1+r2),p3) = 0;
    p4.1 = 1 & p4.2 = 1 & p4.3 = 1 by A4,FINSEQ_1:45;
    hence qfconic(0,0,0,r1,r2,-(r1+r2),p4) = 0;
    p5.1 = e1 & p5.2 = e2 & p5.3 = e3 by A5,FINSEQ_1:45;
    hence r1 * e1 * e2 + r2 * e1 * e3 = (r1 + r2) * e2 * e3 by A8;
    p6.1 = f1 & p6.2 = f2 & p6.3 = f3 by A6,FINSEQ_1:45;
    hence r1 * f1 * f2 + r2 * f1 * f3 = (r1 + r2) * f2 * f3 by A9;
  end;
