
theorem Th57:
  for P,Q being Element of BK_model st P <> Q holds
  ex R being Element of absolute st
  (for p,q,r being non point_at_infty Element of ProjectiveSpace TOP-REAL 3 st
  p = P & q = Q & r = R holds between RP3_to_T2 q,RP3_to_T2 p,RP3_to_T2 r)
  proof
    let P,Q be Element of BK_model;
    assume
A1: P <> Q;
    then consider P1,P2 be Element of absolute such that
A2: P1 <> P2 and
A3: P,Q,P1 are_collinear & P,Q,P2 are_collinear by BKMODEL2:20;
    reconsider p = P,q = Q,p1 = P1,
               p2 = P2 as Element of ProjectiveSpace TOP-REAL 3;
    now
      consider u be Element of TOP-REAL 3 such that
A4:   u is not zero and
A5:   p = Dir u by ANPROJ_1:26;
      u.3 <> 0 by A4,A5,BKMODEL2:2;
      then u`3 <> 0 by EUCLID_5:def 3;
      hence p is non point_at_infty by A4,A5,Th40;
      consider v be Element of TOP-REAL 3 such that
A6:   v is not zero and
A7:   q = Dir v by ANPROJ_1:26;
      v.3 <> 0 by A6,A7,BKMODEL2:2;
      then v`3 <> 0 by EUCLID_5:def 3;
      hence q is non point_at_infty by A6,A7,Th40;
      consider w be non zero Element of TOP-REAL 3 such that
A8:   w.3 = 1 and
A9:   p1 = Dir w by BKMODEL3:30;
      w`3 <> 0 by A8,EUCLID_5:def 3;
      hence p1 is non point_at_infty by A9,Th40;
      consider x be non zero Element of TOP-REAL 3 such that
A10:  x.3 = 1 and
A11:  p2 = Dir x by BKMODEL3:30;
      x`3 <> 0 by A10,EUCLID_5:def 3;
      hence p2 is non point_at_infty by A11,Th40;
    end;
    then reconsider p,q,p1,p2 as
      non point_at_infty Element of ProjectiveSpace TOP-REAL 3;
    per cases by A1,A2,A3,Th56;
    suppose
A12:  between RP3_to_T2 q,RP3_to_T2 p,RP3_to_T2 p1;
      take P1;
      thus thesis by A12;
    end;
    suppose
A13:  between RP3_to_T2 q,RP3_to_T2 p,RP3_to_T2 p2;
      take P2;
      thus thesis by A13;
    end;
  end;
