
theorem Th50:
  for P,Q being non point_at_infty Element of ProjectiveSpace TOP-REAL 3 st
  RP3_to_T2 P = RP3_to_T2 Q holds P = Q
  proof
    let P,Q be non point_at_infty Element of ProjectiveSpace TOP-REAL 3;
    assume
A1: RP3_to_T2 P = RP3_to_T2 Q;
    consider u be non zero Element of TOP-REAL 3 such that
A2: P = Dir u and
A3: u`3 = 1 and
A4: RP3_to_REAL2 P = |[u`1,u`2]| by Def05;
    consider v be non zero Element of TOP-REAL 3 such that
A5: Q = Dir v and
A6: v`3 = 1 and
A7: RP3_to_REAL2 Q = |[v`1,v`2]| by Def05;
    v`1 = u`1 & v`2 = u`2 & v`3 = u`3 by A1,A4,A7,A3,A6,FINSEQ_1:77;
    then |[u`1,u`2,u`3]| = v by EUCLID_5:3;
    hence thesis by A2,A5,EUCLID_5:3;
  end;
