
theorem Th20:
  for P1,P2 being Element of absolute st pole_infty P1 = pole_infty P2
  holds P1 = P2 or (ex u being non zero Element of TOP-REAL 3 st
  P1 = Dir u & P2 = Dir |[-u`1,-u`2,1]| & u`3 = 1)
  proof
    let P1,P2 be Element of absolute;
    assume
A1: pole_infty P1 = pole_infty P2;
    consider u1 be non zero Element of TOP-REAL 3 such that
A2: P1 = Dir u1 & u1.3 = 1 & (u1.1)^2 + (u1.2)^2 = 1 &
    pole_infty P1 = Dir |[- u1.2,u1.1,0]| by Def03;
    consider u2 be non zero Element of TOP-REAL 3 such that
A3: P2 = Dir u2 & u2.3 = 1 & (u2.1)^2 + (u2.2)^2 = 1 &
    pole_infty P2 = Dir |[- u2.2,u2.1,0]| by Def03;
    reconsider w1 = |[- u1.2,u1.1,0]| as non zero Element of TOP-REAL 3
      by A2,BKMODEL1:91;
    reconsider w2 = |[- u2.2,u2.1,0]| as non zero Element of TOP-REAL 3
      by A3,BKMODEL1:91;
    are_Prop w1,w2 by A1,A2,A3,ANPROJ_1:22;
    then consider a be Real such that
    a <> 0 and
A5: w1 = a * w2 by ANPROJ_1:1;
    a * w2 = |[ a * (- u2.2), a * u2.1, a * 0 ]| by EUCLID_5:8; then
A6: - u1.2 = a * (- u2.2) & u1.1 = a * u2.1 by A5,FINSEQ_1:78; then
A7: 1 = (a * u2.1) * (a * u2.1) + (a * u2.2)^2 by A2
     .= a * a * (u2.1 * u2.1 + u2.2 * u2.2)
     .= a^2 by A3;
A8: a = 1 implies P1 = P2
    proof
      assume a = 1;
      then u1`1 = u2.1 & u1`2 = u2.2 & u1`3 = u2.3
        by A2,A3,A6,EUCLID_5:def 1,def 2,def 3; then
A9:   u1`1 = u2`1 & u1`2 = u2`2 & u1`3 = u2`3 by EUCLID_5:def 1,def 2,def 3;
      u1 = |[u1`1,u1`2,u1`3]| by EUCLID_5:3
        .= u2 by A9,EUCLID_5:3;
      hence thesis by A2,A3;
    end;
    a = -1 implies ex u being non zero Element of TOP-REAL 3 st
    P1 = Dir u & P2 = Dir |[-u`1,-u`2,1]| & u`3 = 1
    proof
      assume a = -1;
      then u1`1 = - u2.1 & u1`2 = - u2.2 & u2`3 = 1
        by A3,A6,EUCLID_5:def 1,def 2,def 3; then
A10:   u1`1 = - u2`1 & u1`2 = - u2`2 & u2`3 = 1 by EUCLID_5:def 1,def 2;
      take u1;
      thus thesis by A10,EUCLID_5:3,A3,A2,EUCLID_5:def 3;
    end;
    hence thesis by A7,A8,SQUARE_1:41;
  end;
