
theorem Th47:
  for P,Q,P1 being non point_at_infty Element of ProjectiveSpace TOP-REAL 3 st
  P in BK_model & Q in BK_model & P1 in absolute :::& P,Q,P1 are_collinear
    holds
  not between RP3_to_T2 Q,RP3_to_T2 P1,RP3_to_T2 P
  proof
    let P,Q,P1 be non point_at_infty Element of ProjectiveSpace TOP-REAL 3;
    assume
A1: P in BK_model & Q in BK_model & P1 in absolute;
    set P9 = RP3_to_T2 P, Q9 = RP3_to_T2 Q, P19 = RP3_to_T2 P1;
    assume
A2: between Q9,P19,P9;
    consider u be non zero Element of TOP-REAL 3 such that
A3: P = Dir u & u`3 = 1 & RP3_to_REAL2 P = |[u`1,u`2]| by Def05;
    consider v be non zero Element of TOP-REAL 3 such that
A4: Q = Dir v & v`3 = 1 & RP3_to_REAL2 Q = |[v`1,v`2]| by Def05;
    consider w1 be non zero Element of TOP-REAL 3 such that
A5: P1 = Dir w1 & w1`3 = 1 & RP3_to_REAL2 P1 = |[w1`1,w1`2]| by Def05;
A6: Tn2TR P19 in LSeg(Tn2TR Q9,Tn2TR P9) by A2,GTARSKI2:20;
    reconsider u9 = Tn2TR P19, v9 = Tn2TR Q9,
               w9 = Tn2TR P9 as Element of TOP-REAL 2;
    |[u9`1,u9`2]| = |[w1`1,w1`2]| by A5,EUCLID:53;
    then
A7: u9`1 = w1`1 & u9`2 = w1`2 by FINSEQ_1:77;
    |[v9`1,v9`2]| = |[v`1,v`2]| by A4,EUCLID:53;
    then
A8: v9`1 = v`1 & v9`2 = v`2 by FINSEQ_1:77;
    |[w9`1,w9`2]| = |[u`1,u`2]| by A3,EUCLID:53;
    then
A9: w9`1 = u`1 & w9`2 = u`2 by FINSEQ_1:77;
    reconsider pu = |[u9`1,u9`2,1]|, pv = |[v9`1,v9`2,1]|,
               pw = |[w9`1,w9`2,1]| as non zero Element of TOP-REAL 3;
    pu in LSeg(pw,pv) by A6,Th45;
    then consider r be Real such that
A10: 0 <= r & r <= 1 and
A11: pu = (1 - r) * pw + r * pv by RLTOPSP1:76;
    now
      thus Q = Dir pv by A4,A8,EUCLID_5:3;
      thus P = Dir pw by A3,A9,EUCLID_5:3;
      thus P1 = Dir pu by A5,A7,EUCLID_5:3;
      thus pv.3 = pv`3 by EUCLID_5:def 3
               .= 1 by EUCLID_5:2;
      thus pw.3 = pw`3 by EUCLID_5:def 3
               .= 1 by EUCLID_5:2;
      thus pu = r * pv + (1 - r) * pw by A11;
    end;
    then P1 is Element of BK_model by A1,A10,Th17;
    hence contradiction by A1,BKMODEL2:1,XBOOLE_0:def 4;
  end;
