theorem
  for P being Element of absolute holds
  ex u st (u.1)^2 + (u.2)^2 = 1 & u.3 = 1 & P = Dir u
  proof
    let P be Element of absolute;
    consider u be Element of TOP-REAL 3 such that
A1: u is non zero and
A2: Dir u = P by ANPROJ_1:26;
    reconsider v1 = u.1/u.3,v2 = u.2/u.3 as Real;
    reconsider v = |[v1,v2,1]| as Element of TOP-REAL 3;
A3: v.3 = v`3 by EUCLID_5:def 3
       .= 1 by EUCLID_5:2;
A4: v is non zero by EUCLID_5:4,FINSEQ_1:78;
    u.3 * v = |[ u.3 * v1, u.3 * v2, u.3 * 1]| by EUCLID_5:8
           .= |[u.1,u.3 * v2,u.3]| by A1,A2,Th67,XCMPLX_1:87
           .= |[u.1,u.2,u.3]| by A1,A2,Th67,XCMPLX_1:87
           .= |[u`1,u.2,u.3]| by EUCLID_5:def 1
           .= |[u`1,u`2,u.3]| by EUCLID_5:def 2
           .= |[u`1,u`2,u`3]| by EUCLID_5:def 3
           .= u by EUCLID_5:3;
    then are_Prop u,v by A1,A2,Th67,ANPROJ_1:1;
    then
A5: Dir v = Dir u by A1,A4,ANPROJ_1:22;
    |[v.1,v.2]| in circle(0,0,1) by A2,A3,A4,A5,Th68;
    then (v.1)^2 + (v.2)^2 = 1^2 by Th12
                          .= 1;
    hence thesis by A2,A3,A4,A5;
  end;
