
theorem Th51:
  for R1,R2 being non point_at_infty Element of ProjectiveSpace TOP-REAL 3
  st R1 in absolute & R2 in absolute & R1 <> R2 holds
  ex P being Element of BK-model-Plane st
  between RP3_to_T2 R1,BK_to_T2 P,RP3_to_T2 R2
  proof
    let R1,R2 be non point_at_infty Element of ProjectiveSpace TOP-REAL 3;
    assume that
A1: R1 in absolute & R2 in absolute and
A2: R1 <> R2;
    consider u1 be non zero Element of TOP-REAL 3 such that
A3: R1 = Dir u1 and
A4: u1`3 = 1 and
A5: RP3_to_REAL2 R1 = |[u1`1,u1`2]| by Def05;
    u1.3 = 1 by A4,EUCLID_5:def 3;
    then |[u1.1,u1.2]| in circle(0,0,1) by A1,A3,BKMODEL1:84;
    then
A6: |[u1`1,u1.2]| in circle(0,0,1) by EUCLID_5:def 1;
    consider u2 be non zero Element of TOP-REAL 3 such that
A7: R2 = Dir u2 and
A8: u2`3 = 1 and
A9: RP3_to_REAL2 R2 = |[u2`1,u2`2]| by Def05;
    u2.3 = 1 by A8,EUCLID_5:def 3;
    then |[u2.1,u2.2]| in circle(0,0,1) by A1,A7,BKMODEL1:84;
    then
A10: |[u2`1,u2.2]| in circle(0,0,1) by EUCLID_5:def 1;
    reconsider P1 = RP3_to_T2 R1,
    P2 = RP3_to_T2 R2 as Point of TarskiEuclid2Space;
    Tn2TR P1 in circle(0,0,1) & Tn2TR P2 in circle(0,0,1) & P1 <> P2
      by A2,Th50,A6,A5,A9,A10,EUCLID_5:def 2;
    hence thesis by Th49;
  end;
