
theorem
  for P,Q,R being Element of ProjectiveSpace TOP-REAL 3
  for u,v,w being non zero Element of TOP-REAL 3 st
  P = Dir u & Q = Dir v & R = Dir w & u`3 <> 0 & v`3 = 0 &
  w = |[u`1 + v`1,u`2 + v`2, u`3]| holds R <> P & R <> Q
  proof
    let P,Q,R be Element of ProjectiveSpace TOP-REAL 3;
    let u,v,w be non zero Element of TOP-REAL 3;
    assume that
A1: P = Dir u and
A2: Q = Dir v and
A3: R = Dir w and
A4: u`3 <> 0 and
A5: v`3 = 0 and
A6: w = |[u`1 + v`1,u`2 + v`2, u`3]|;
    hereby
      assume R = P;
      then are_Prop u,w by A1,A3,ANPROJ_1:22;
      then consider a be Real such that a <> 0 and
A7:   u = a * w by ANPROJ_1:1;
A8:   |[u`1,u`2,u`3]| = u by EUCLID_5:3
                     .= |[a * w`1,a * w`2,a * w`3]| by A7,EUCLID_5:7;
      then |[u`1,u`2,u`3]| = |[a * w`1,a * w`2, a * u`3]| by A6,EUCLID_5:2;
      then u`3 = a * u`3 by FINSEQ_1:78;
      then
A9:   a = 1 by A4,XCMPLX_1:7;
      w`1 = u`1 + v`1 & w`2 = u`2 + v`2 & w`3 = u`3 by A6,EUCLID_5:2;
      then u`1 = u`1 + v`1 & u`2 = u`2 + v`2 by A8,A9,FINSEQ_1:78;
      hence contradiction by A5,EUCLID_5:3,4;
    end;
    hereby
      assume R = Q;
      then are_Prop v,w by A2,A3,ANPROJ_1:22;
      then consider b be Real such that
A11:  b <> 0 and
A12:  v = b * w by ANPROJ_1:1;
      |[v`1,v`2,v`3]| = v by EUCLID_5:3
                     .= |[b * w`1,b * w`2,b * w`3]| by A12,EUCLID_5:7;
      then |[v`1,v`2,v`3]| = |[b * w`1,b * w`2, b * u`3]| by A6,EUCLID_5:2;
      hence contradiction by A4,A11,A5,FINSEQ_1:78;
    end;
  end;
