
theorem Th49:
  for R1,R2 being Point of TarskiEuclid2Space st
  Tn2TR R1 in circle(0,0,1) & Tn2TR R2 in circle(0,0,1) & R1 <> R2 holds
  ex P being Element of BK-model-Plane st between R1,BK_to_T2 P,R2
  proof
    let R1,R2 be Point of TarskiEuclid2Space;
    assume
A1: Tn2TR R1 in circle(0,0,1) & Tn2TR R2 in circle(0,0,1) & R1 <> R2;
    reconsider P = Tn2TR R1, Q = Tn2TR R2 as Element of TOP-REAL 2;
A2: P = |[P`1,P`2]| & Q = |[Q`1,Q`2]| by EUCLID:53;
    reconsider w = |[ (P`1+Q`1)/2,(P`2+Q`2)/2 ]| as Element of TOP-REAL 2;
    reconsider u9 = |[P`1,P`2,1]|,
               v9 = |[Q`1,Q`2,1]| as non zero Element of TOP-REAL 3;
    reconsider w9 = |[w`1,w`2,1]| as non zero Element of TOP-REAL 3;
    reconsider P9 = Dir u9, Q9 = Dir v9,
               R9 = Dir w9 as Point of ProjectiveSpace TOP-REAL 3
                 by ANPROJ_1:26;
    u9`3 = 1 & v9`3 = 1 by EUCLID_5:2;
    then reconsider P9,Q9 as
      non point_at_infty Point of ProjectiveSpace TOP-REAL 3 by Th40;
    w9`3 <> 0 by EUCLID_5:2;
    then reconsider R9 as
      non point_at_infty Point of ProjectiveSpace TOP-REAL 3 by Th40;
    reconsider R99 = RP3_to_T2 R9 as Point of TarskiEuclid2Space;
    consider w99 be non zero Element of TOP-REAL 3 such that
A3: R9 = Dir w99 & w99`3 = 1 & RP3_to_REAL2 R9 = |[w99`1,w99`2]|  by Def05;
A4: w9`1 = w`1 & w9`2 = w`2 by EUCLID_5:2;
    w99.3 = 1 & w9`3 = 1 by A3,EUCLID_5:2,def 3;
    then w99.3 = w9.3 & w9.3 <> 0 by EUCLID_5:def 3;
    then
A5: w99 = w9 by A3,BKMODEL1:43;
A6: Tn2TR R99 = w by A3,A5,A4,EUCLID:53;
    w = |[ P`1 / 2 + Q`1 / 2, P`2 / 2 + Q`2 / 2]|
     .= |[ P`1/2 ,P`2/2]| + |[Q`1/2,Q`2/2]| by EUCLID:56
     .= 1/2 * |[P`1,P`2]| + |[Q`1/2,Q`2/2]| by EUCLID:58
     .= (1 - 1/2) * P + 1/2 * Q by A2,EUCLID:58;
    then w in {(1-r)*P + r * Q where r is Real: 0 <= r & r <= 1};
    then
A7: w in LSeg(Tn2TR R1,Tn2TR R2) by RLTOPSP1:def 2;
    now
      now
        thus P9 = Dir u9;
        u9`3 = 1 by EUCLID_5:2;
        hence u9.3 = 1 by EUCLID_5:def 3;
        thus |[P`1,P`2]| in circle(0,0,1) by A1,EUCLID:53;
        u9`1 = P`1 & u9`2 = P`2 by EUCLID_5:2;
        then P`1 = u9.1 & P`2 = u9.2 by EUCLID_5:def 1,def 2;
        hence |[u9.1,u9.2]| in circle(0,0,1) by A1,EUCLID:53;
      end;
      then P9 is Element of absolute by BKMODEL1:86;
      hence P9 in absolute;
      now
        thus Q9 = Dir v9;
        v9`3 = 1 by EUCLID_5:2;
        hence v9.3 = 1 by EUCLID_5:def 3;
        v9`1 = Q`1 & v9`2 = Q`2 by EUCLID_5:2;
        then Q`1 = v9.1 & Q`2 = v9.2 by EUCLID_5:def 1,def 2;
        hence |[v9.1,v9.2]| in circle(0,0,1) by A1,EUCLID:53;
      end;
      then Q9 is Element of absolute by BKMODEL1:86;
      hence Q9 in absolute;
      thus P9 <> Q9
      proof
        assume
A8:     P9 = Q9;
        now
          thus Dir u9 = Dir v9 by A8;
          thus u9.3 = u9`3 by EUCLID_5:def 3
                   .= 1 by EUCLID_5:2;
          thus v9.3 = v9`3 by EUCLID_5:def 3
                   .= 1 by EUCLID_5:2;
        end;
        then u9 = v9 by BKMODEL1:43;
        then P`1 = Q`1 & P`2 = Q`2 by FINSEQ_1:78;
        then |[P`1,P`2]| = Q by EUCLID:53;
        hence contradiction by A1,EUCLID:53;
      end;
      thus u9`3 = 1 & v9`3 = 1 by EUCLID_5:2;
      thus w9 = |[(u9`1+ v9`1)/2,(u9`2+v9`2)/2,1]|
      proof
        u9`1 = P`1 & v9`1 = Q`1 & u9`2 = P`2 & v9`2 = Q`2 &
          w`1 = (P`1+Q`1)/2 & w`2 = (P`2+Q`2)/2 by EUCLID:52,EUCLID_5:2;
        hence thesis;
      end;
    end;
    then reconsider AR9 = R9 as Element of BK-model-Plane by Th48;
    consider r be Element of BK_model such that
A9: AR9 = r and
A10: BK_to_T2 AR9 = BK_to_REAL2 r by Def01;
    take AR9;
    now
      thus R99 = RP3_to_T2 R9;
      consider x be non zero Element of TOP-REAL 3 such that
A11:  Dir x = r & x.3 = 1 & BK_to_REAL2 r = |[x.1,x.2]| by BKMODEL2:def 2;
      now
        thus Dir x = Dir w9 by A11,A9;
        thus x.3 <> 0 by A11;
        w9.3 = w9`3 by EUCLID_5:def 3
            .= 1 by EUCLID_5:2;
        hence x.3 = w9.3 by A11;
      end;
      then x = w9 by Th16;
      then BK_to_REAL2 r = |[w9`1,w9.2]| by A11,EUCLID_5:def 1
                        .= |[w`1,w`2]| by A4,EUCLID_5:def 2
                        .= w by EUCLID:53;
      hence RP3_to_T2 R9 = BK_to_T2 AR9 by A3,A5,A4,EUCLID:53,A10;
    end;
    hence
    between R1,BK_to_T2 AR9,R2 by A7,A6,GTARSKI2:20;
  end;
