
theorem Th53:
  for P being non point_at_infty Element of ProjectiveSpace TOP-REAL 3
  st P in absolute holds RP3_to_REAL2 P in circle(0,0,1)
  proof
    let P be non point_at_infty Element of ProjectiveSpace TOP-REAL 3;
    assume
A1: P in absolute;
    consider u be non zero Element of TOP-REAL 3 such that
A2: P = Dir u & u`3 = 1 & RP3_to_REAL2 P = |[u`1,u`2]| by Def05;
    u.3 = 1 by A2,EUCLID_5:def 3;
    then |[u.1,u.2]| in circle(0,0,1) by A1,A2,BKMODEL1:84;
    then |[u`1,u.2]| in circle(0,0,1) by EUCLID_5:def 1;
    hence thesis by A2,EUCLID_5:def 2;
  end;
