
theorem Th19:
  for P being Element of absolute holds P <> pole_infty P
  proof
    let P be Element of absolute;
    assume
A1: P = pole_infty P;
    consider u being non zero Element of TOP-REAL 3 such that
A2: P = Dir u & u.3 = 1 & (u.1)^2 + (u.2)^2 = 1 &
      pole_infty P = Dir |[- u.2,u.1,0]| by Def03;
A3: |[-u.2,u.1,0]| is non zero by A2,BKMODEL1:91;
    are_Prop u,|[-u.2,u.1,0]| by A1,A2,A3,ANPROJ_1:22;
    then consider a be Real such that
    a <> 0 and
A4: u = a * |[-u.2,u.1,0]| by ANPROJ_1:1;
    1 = a * (|[-u.2,u.1,0]|).3 by A2,A4,RVSUM_1:44
     .= a * (|[-u.2,u.1,0]|)`3 by EUCLID_5:def 3
     .= a * 0 by EUCLID_5:2
     .= 0;
    hence contradiction;
  end;
