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;

theorem Th10:
  Dir u1 = Dir u2 & qfconic(a,b,c,d,e,f,u1) = 0 implies
    qfconic(a,b,c,d,e,f,u2) = 0
  proof
    assume that
A1: Dir u1 = Dir u2 and
A2: qfconic(a,b,c,d,e,f,u1) = 0;
    are_Prop u1,u2 by A1,ANPROJ_1:22;
    then consider r be Real such that
A3: r <> 0 and
A4: u1 = r * u2 by ANPROJ_1:1;
    r is non zero by A3; then
A5: r * r <> 0 by ORDINAL1:def 14;
    u1.1 = r * u2.1 & u1.2 = r * u2.2 & u1.3 = r * u2.3 by A4,RVSUM_1:44;
    then r * r * (a * u2.1 * u2.1 + b * u2.2 * u2.2 + c * u2.3 * u2.3
      + d * u2.1 * u2.2 + e * u2.1 * u2.3 + f * u2.3 * u2.2) = r * r * 0
      by A2;
    hence thesis by A5,XCMPLX_1:5;
  end;
