
theorem Th55:
  for P,Q,R1,R2 being non point_at_infty Element of ProjectiveSpace
  TOP-REAL 3 st P <> Q & P in BK_model & R1 in absolute &
  R2 in absolute & between RP3_to_T2 P,RP3_to_T2 Q,RP3_to_T2 R1 &
  between RP3_to_T2 P,RP3_to_T2 Q,RP3_to_T2 R2 holds R1 = R2
  proof
    let P,Q,R1,R2 be non point_at_infty Element of ProjectiveSpace TOP-REAL 3;
    assume that
A1: P <> Q and
A2: P in BK_model and
A3: R1 in absolute and
A4: R2 in absolute and
A5: between RP3_to_T2 P,RP3_to_T2 Q,RP3_to_T2 R1 and
A6: between RP3_to_T2 P,RP3_to_T2 Q,RP3_to_T2 R2;
    assume R1 <> R2;
    then consider S be Element of BK-model-Plane such that
A7: between RP3_to_T2 R1,BK_to_T2 S,RP3_to_T2 R2 by A3,A4,Th51;
    set p = RP3_to_T2 P, q = RP3_to_T2 Q, r1 = RP3_to_T2 R1,
      r2 = RP3_to_T2 R2, s = BK_to_T2 S;
    (between p,r1,r2 or between p,r2,r1) & between r1,s,r2 & between r2,s,r1
      by A7,GTARSKI1:16,A1,A5,A6,Th50,GTARSKI3:56;
    then per cases by GTARSKI1:19;
    suppose
A8:   between p,r1,s;
A9:   RP3_to_REAL2 R1 in circle(0,0,1) by A3,Th53;
      now
        thus between p,r1,s by A8;
        reconsider P9 = P as Element of BK_model by A2;
        BK_to_REAL2 P9 = RP3_to_REAL2 P by Th54;
        hence Tn2TR p in inside_of_circle(0,0,1);
        thus Tn2TR s in inside_of_circle(0,0,1) by Th02;
      end;
      then Tn2TR r1 in inside_of_circle(0,0,1) by Th52;
      then RP3_to_REAL2 R1 in inside_of_circle(0,0,1) /\ circle(0,0,1)
        by A9,XBOOLE_0:def 4;
      hence contradiction by XBOOLE_0:def 7,TOPREAL9:54;
    end;
    suppose
A10:  between p,r2,s;
A11:  RP3_to_REAL2 R2 in circle(0,0,1) by A4,Th53;
      now
        thus between p,r2,s by A10;
        reconsider P9 = P as Element of BK_model by A2;
        reconsider P99 = P9 as POINT of BK-model-Plane;
        BK_to_REAL2 P9 = RP3_to_REAL2 P by Th54;
        hence Tn2TR p in inside_of_circle(0,0,1);
        thus Tn2TR s in inside_of_circle(0,0,1) by Th02;
      end;
      then Tn2TR r2 in inside_of_circle(0,0,1) by Th52;
      then RP3_to_REAL2 R2 in inside_of_circle(0,0,1) /\ circle(0,0,1)
        by A11,XBOOLE_0:def 4;
      hence contradiction by XBOOLE_0:def 7,TOPREAL9:54;
    end;
  end;
