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 Th37:
  Dir u = Dir v & u.3 = v.3 & v.3 <> 0 implies u = v
  proof
    assume that
A1: Dir u = Dir v and
A2: u.3 = v.3 and
A3: v.3 <> 0;
    are_Prop u,v by A1,ANPROJ_1:22;
    then consider a be Real such that
    a <> 0 and
A4: u = a * v by ANPROJ_1:1;
    reconsider b = 1 - a, c = v.3 as Real;
A5: |[u`1,u`2,u`3]| = a * v by A4,EUCLID_5:3
                   .= |[ a * v`1,a*v`2,a*v`3]| by EUCLID_5:7;
    v.3 = u`3 by A2,EUCLID_5:def 3
       .= a*v`3 by A5,FINSEQ_1:78
       .= a * v.3 by EUCLID_5:def 3;
    then (1 - a) * v.3 = 0 & c = v.3;
    then b = 0 by A3;
    then u = |[ 1 * v`1,1 * v`2, 1 * v`3]| by A4,EUCLID_5:7
          .= v by EUCLID_5:3;
    hence thesis;
  end;
