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;

theorem Th18:
  (qfconic(a,b,c,d,e,f,|[1,0,0]|) = 0 implies a = 0) &
  (qfconic(a,b,c,d,e,f,|[0,1,0]|) = 0 implies b = 0) &
  (qfconic(a,b,c,d,e,f,|[0,0,1]|) = 0 implies c = 0) &
  (qfconic(0,0,0,d,e,f,|[1,1,1]|) = 0 implies d + e + f = 0) 
  proof
    hereby
      assume
A1:   qfconic(a,b,c,d,e,f,|[1,0,0]|) = 0;
      |[1,0,0]|.1 = |[1,0,0]|`1 &
      |[1,0,0]|.2 = |[1,0,0]|`2 &
      |[1,0,0]|.3 = |[1,0,0]|`3
        by EUCLID_5:def 1,EUCLID_5:def 2,EUCLID_5:def 3;
      then |[1,0,0]|.1 = 1 & |[1,0,0]|.2 = 0 & |[1,0,0]|.3 = 0 by EUCLID_5:2;
      hence a = 0 by A1;
    end;
    hereby
      assume
A2:   qfconic(a,b,c,d,e,f,|[0,1,0]|) = 0;
      |[0,1,0]|.1 = |[0,1,0]|`1 &
      |[0,1,0]|.2 = |[0,1,0]|`2 &
      |[0,1,0]|.3 = |[0,1,0]|`3
        by EUCLID_5:def 1,EUCLID_5:def 2,EUCLID_5:def 3;
      then |[0,1,0]|.1 = 0 & |[0,1,0]|.2 = 1 & |[0,1,0]|.3 = 0 by EUCLID_5:2;
      hence b = 0 by A2;
    end;
    hereby
      assume
A3:   qfconic(a,b,c,d,e,f,|[0,0,1]|) = 0;
      |[0,0,1]|.1 = |[0,0,1]|`1 &
      |[0,0,1]|.2 = |[0,0,1]|`2 &
      |[0,0,1]|.3 = |[0,0,1]|`3
        by EUCLID_5:def 1,EUCLID_5:def 2,EUCLID_5:def 3;
      then |[0,0,1]|.1 = 0 & |[0,0,1]|.2 = 0 & |[0,0,1]|.3 = 1 by EUCLID_5:2;
      hence c = 0 by A3;
    end;
    hereby
      assume
A4:   qfconic(0,0,0,d,e,f,|[1,1,1]|) = 0;
      |[1,1,1]|.1 = |[1,1,1]|`1 &
      |[1,1,1]|.2 = |[1,1,1]|`2 &
      |[1,1,1]|.3 = |[1,1,1]|`3
        by EUCLID_5:def 1,EUCLID_5:def 2,EUCLID_5:def 3;
      then |[1,1,1]|.1 = 1 & |[1,1,1]|.2 = 1 & |[1,1,1]|.3 = 1 by EUCLID_5:2;
      hence d + e + f = 0 by A4;
    end;
  end;
