reserve a,b,c,d,e,f for Real,
        g           for positive Real,
        x,y         for Complex,
        S,T         for Element of REAL 2,
        u,v,w       for Element of TOP-REAL 3;
reserve a,b,c for Element of F_Real,
          M,N for Matrix of 3,F_Real;
reserve D        for non empty set;
reserve d1,d2,d3 for Element of D;
reserve A        for Matrix of 1,3,D;
reserve B        for Matrix of 3,1,D;
reserve u,v for non zero Element of TOP-REAL 3;

theorem
  not Dir101,Dirm101,Dir011 are_collinear &
  not Dir101,Dirm101,Dir010 are_collinear &
  not Dir101,Dir011,Dir010 are_collinear &
  not Dirm101,Dir011,Dir010 are_collinear
  proof
    thus not Dir101,Dirm101,Dir011 are_collinear
    proof
      assume
A1:   Dir101,Dirm101,Dir011 are_collinear;
      set p = |[1,0,1]|, q = |[-1,0,1]|, r = |[0,1,1]|;
A2:   p`1 = 1 & p`2 = 0 & p`3 = 1 & q`1 = -1 & q`2 = 0 & q`3 = 1 &
      r`1 = 0 & r`2 = 1 & r`3 = 1 by EUCLID_5:2;
      p is non zero & q is non zero & r is non zero by EUCLID_5:4,FINSEQ_1:78;
       then 0 = |{p,q,r}| by A1,Th01
            .= 1 * 0 * 1 - 1 * 0 * 0 - 1 * 1 * 1 + 0 * 1 * 0
                - 0 * (-1) * 1 + 1 * (-1) * 1 by A2,ANPROJ_8:27
            .= -2;
      hence contradiction;
    end;
    thus not Dir101,Dirm101,Dir010 are_collinear
    proof
      assume
A3:   Dir101,Dirm101,Dir010 are_collinear;
      set p = |[1,0,1]|, q = |[-1,0,1]|, r = |[0,1,0]|;
A4:   p`1 = 1 & p`2 = 0 & p`3 = 1 & q`1 = -1 & q`2 = 0 & q`3 = 1 &
      r`1 = 0 & r`2 = 1 & r`3 = 0 by EUCLID_5:2;
      p is non zero & q is non zero & r is non zero by EUCLID_5:4,FINSEQ_1:78;
      then 0 = |{p,q,r}| by A3,ANPROJ_9:def 6,Th01
            .= 1 * 0 * 0 - 1 * 0 * 0 - 1 * 1 * 1 + 0 * 1 * 0
                - 0 * (-1) * 0 + 1 * (-1) * 1 by A4,ANPROJ_8:27
            .= -2;
      hence contradiction;
    end;
    thus not Dir101,Dir011,Dir010 are_collinear
    proof
      assume
A5:   Dir101,Dir011,Dir010 are_collinear;
      set p = |[1,0,1]|, q = |[0,1,1]|, r = |[0,1,0]|;
A6:   p`1 = 1 & p`2 = 0 & p`3 = 1 & q`1 = 0 & q`2 = 1 & q`3 = 1 &
      r`1 = 0 & r`2 = 1 & r`3 = 0 by EUCLID_5:2;
      p is non zero & q is non zero & r is non zero by EUCLID_5:4,FINSEQ_1:78;
      then 0 = |{p,q,r}| by A5,ANPROJ_9:def 6,Th01
      .= p`1 * q`2 * r`3 - p`3*q`2*r`1 - p`1*q`3*r`2 + p`2*q`3*r`1 -
                p`2*q`1*r`3 + p`3*q`1*r`2 by ANPROJ_8:27
            .= -1 by A6;
      hence contradiction;
    end;
    thus not Dirm101,Dir011,Dir010 are_collinear
    proof
      assume
A7:   Dirm101,Dir011,Dir010 are_collinear;
      set p = |[-1,0,1]|, q = |[0,1,1]|, r = |[0,1,0]|;
A8:   p`1 = -1 & p`2 = 0 & p`3 = 1 & q`1 = 0 & q`2 = 1 & q`3 = 1 &
      r`1 = 0 & r`2 = 1 & r`3 = 0 by EUCLID_5:2;
      p is non zero & q is non zero & r is non zero by EUCLID_5:4,FINSEQ_1:78;
      then 0 = |{p,q,r}| by A7,ANPROJ_9:def 6,Th01
            .= p`1 * q`2 * r`3 - p`3*q`2*r`1 - p`1*q`3*r`2 + p`2*q`3*r`1 -
                p`2*q`1*r`3 + p`3*q`1*r`2 by ANPROJ_8:27
            .= 1 by A8;
      hence contradiction;
    end;
  end;
