 reserve i,n for Nat;
 reserve r for Real;
 reserve ra for Element of F_Real;
 reserve a,b,c for non zero Element of F_Real;
 reserve u,v for Element of TOP-REAL 3;
 reserve p1 for FinSequence of (1-tuples_on REAL);
 reserve pf,uf for FinSequence of F_Real;
 reserve N for Matrix of 3,F_Real;
 reserve K for Field;
 reserve k for Element of K;
 reserve N,N1,N2 for invertible Matrix of 3,F_Real;
 reserve P,P1,P2,P3 for Point of ProjectiveSpace TOP-REAL 3;

theorem
  Dir100 <> Dir010 &
  Dir100 <> Dir001 &
  Dir100 <> Dir111 &
  Dir010 <> Dir001 &
  Dir010 <> Dir111 &
  Dir001 <> Dir111
  proof
    assume
A1: Dir100 = Dir010 or Dir100 = Dir001 or Dir100 = Dir111 or Dir010 = Dir001
      or Dir010 = Dir111 or Dir001 = Dir111;
    consider u100 be Element of TOP-REAL 3 such that
A2: u100 = |[1,0,0]| and
A3: Dir100 = Dir u100;
    consider u010 be Element of TOP-REAL 3 such that
A4: u010 = |[0,1,0]| and
A5: Dir010 = Dir u010;
    consider u001 be Element of TOP-REAL 3 such that
A6: u001 = |[0,0,1]| and
A7: Dir001 = Dir u001;
    consider u111 be Element of TOP-REAL 3 such that
A8: u111 = |[1,1,1]| and
A9: Dir111 = Dir u111;
    per cases by A1;
    suppose
A10: Dir100 = Dir010;
      u100 is not zero & u010 is not zero by A2,A4,EUCLID_5:4,FINSEQ_1:78;
      then are_Prop u100, u010 by A3,A5,A10,ANPROJ_1:22;
      then consider a be Real such that
      a <> 0 and
A12:  u100 = a * u010 by ANPROJ_1:1;
      |[1,0,0]| = |[a * 0,a * 1 ,0]| by A2,A4,A12,EUCLID_5:8
               .= |[0,a,0]|;
      then 1 = |[0,a,0]|`1 by EUCLID_5:2;
      hence thesis by EUCLID_5:2;
    end;
    suppose
A13:  Dir100 = Dir001;
      u100 is not zero & u001 is not zero by A2,A6,EUCLID_5:4,FINSEQ_1:78;
      then are_Prop u100, u001 by A13,A3,A7,ANPROJ_1:22;
      then consider a be Real such that
      a <> 0 and
A15:  u100 = a * u001 by ANPROJ_1:1;
      |[1,0,0]| = |[a * 0,a * 0 ,a * 1]| by A2,A6,A15,EUCLID_5:8
               .= |[0,0,a]|;
      then 1 = |[0,0,a]|`1 by EUCLID_5:2;
      hence thesis by EUCLID_5:2;
    end;
    suppose
A16:  Dir100 = Dir111;
      u100 is not zero & u111 is not zero by A2,A8,EUCLID_5:4,FINSEQ_1:78;
      then are_Prop u100, u111 by A16,A3,A9,ANPROJ_1:22;
      then consider a be Real such that
A17:  a <> 0 and
A18:  u100 = a * u111 by ANPROJ_1:1;
      |[1,0,0]| = |[a * 1,a * 1 ,a * 1]| by A2,A8,A18,EUCLID_5:8
               .= |[a,a,a]|;
      then 0 = |[a,a,a]|`2 by EUCLID_5:2;
      hence thesis by A17,EUCLID_5:2;
    end;
    suppose
A19:  Dir010 = Dir001;
      u010 is not zero & u001 is not zero by A4,A6,EUCLID_5:4,FINSEQ_1:78;
      then are_Prop u010, u001 by A5,A7,A19,ANPROJ_1:22;
      then consider a be Real such that
A20:  a <> 0 and
A21:  u010 = a * u001 by ANPROJ_1:1;
      |[0,1,0]| = |[a * 0,a * 0 ,a * 1]| by A4,A6,A21,EUCLID_5:8
               .= |[0,0,a]|;
      then 0 = |[0,0,a]|`3 by EUCLID_5:2;
      hence thesis by A20,EUCLID_5:2;
    end;
    suppose
A22:  Dir010 = Dir111;
      u010 is not zero & u111 is not zero by A4,A8,EUCLID_5:4,FINSEQ_1:78;
      then are_Prop u010, u111 by A22,A5,A9,ANPROJ_1:22;
      then consider a be Real such that
A23:  a <> 0 and
A24:  u010 = a * u111 by ANPROJ_1:1;
      |[0,1,0]| = |[a * 1,a * 1 ,a]| by A4,A8,A24,EUCLID_5:8
               .= |[a,a,a]|;
      then 0 = |[a,a,a]|`1 by EUCLID_5:2;
      hence thesis by A23,EUCLID_5:2;
    end;
    suppose
A25:  Dir001 = Dir111;
      u001 is not zero & u111 is not zero by A6,A8,EUCLID_5:4,FINSEQ_1:78;
      then are_Prop u001, u111 by A7,A9,A25,ANPROJ_1:22;
      then consider a be Real such that
A26:  a <> 0 and
A27:  u001 = a * u111 by ANPROJ_1:1;
      |[0,0,1]| = |[a * 1,a * 1 ,a]| by A6,A8,A27,EUCLID_5:8
               .= |[a,a,a]|;
      then 0 = |[a,a,a]|`1 by EUCLID_5:2;
      hence thesis by A26,EUCLID_5:2;
    end;
  end;
