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;
