
theorem Th48:
  for P,Q,R being non point_at_infty Element of ProjectiveSpace TOP-REAL 3
  for u,v,w being non zero Element of TOP-REAL 3 st
  P in absolute & Q in absolute & P <> Q & P = Dir u & Q = Dir v & R = Dir w &
  u`3 = 1 & v`3 = 1 & w = |[ (u`1 + v`1) / 2,(u`2 + v`2) / 2,1]| holds
  R in BK_model
  proof
    let P,Q,R be non point_at_infty Element of ProjectiveSpace TOP-REAL 3;
    let u,v,w be non zero Element of TOP-REAL 3;
    assume that
A1: P in absolute and
A2: Q in absolute and
A3: P <> Q and
A4: P = Dir u & Q = Dir v & R = Dir w and
A5: u`3 = 1 & v`3 = 1 and
A6: w = |[ (u`1 + v`1) / 2,(u`2 + v`2) / 2, 1]|;
A7: u.3 = 1 & v.3 = 1 by A5,EUCLID_5:def 3;
    reconsider u9 = |[u.1,u.2]|,v9 = |[v.1,v.2]| as Element of TOP-REAL 2;
    set M = the_midpoint_of_the_segment(u9,v9);
A8: w`1 = (u`1+v`1)*1/2 & w`2=(u`2+v`2)*1/2 by A6,EUCLID_5:2;
A9: M = 1/2 * (u9 + v9) by EUCLID12:29
     .= 1/2 * |[u.1 + v.1,u.2 + v.2]| by EUCLID:56
     .= |[(u.1+v.1)*1/2,(u.2+v.2)*1/2]| by EUCLID:58
     .= |[(u`1+v.1)*1/2,(u.2+v.2)*1/2]| by EUCLID_5:def 1
     .= |[(u`1+v.1)*1/2,(u`2+v.2)*1/2]| by EUCLID_5:def 2
     .= |[(u`1+v`1)*1/2,(u`2+v.2)*1/2]| by EUCLID_5:def 1
     .= |[w`1,w`2]| by A8,EUCLID_5:def 2;
    u9 in circle(0,0,1) & v9 in circle(0,0,1) by A7,A1,A2,A4,BKMODEL1:84;
    then
A10: LSeg(u9,v9) \ {u9,v9} c= inside_of_circle(0,0,1) by TOPREAL9:60;
    u9 <> v9
    proof
      assume u9 = v9;
      then u.1 = v.1 & u.2 = v.2 by FINSEQ_1:77;
      then u`1 = v.1 & u`2 = v.2 by EUCLID_5:def 1,def 2;
      then u`1 = v`1 & u`2 = v`2 by EUCLID_5:def 1,def 2;
      then u = |[v`1,v`2,v`3]| by A5,EUCLID_5:3
            .= v by EUCLID_5:3;
      hence contradiction by A4,A3;
    end;
    then M <> u9 & M <> v9 by EUCLID12:32,33;
    then
A11: not M in {u9,v9} by TARSKI:def 2;
    M in LSeg(u9,v9) by EUCLID12:28;
    then M in LSeg(u9,v9) \ {u9,v9} by A11,XBOOLE_0:def 5;
    then reconsider rw = |[w`1,w`2]| as Element of inside_of_circle(0,0,1)
      by A10,A9;
   consider RW2 be Element of TOP-REAL 2 such that
A12: RW2 = rw & REAL2_to_BK rw = Dir |[RW2`1,RW2`2,1]| by BKMODEL2:def 3;
A13: rw`1 = w`1 & rw`2 = w`2 by EUCLID:52;
   |[RW2`1,RW2`2,1]| = |[w`1,w`2,w`3]| by A12,A13,A6,EUCLID_5:2
                    .= w by EUCLID_5:3;
    hence thesis by A12,A4;
  end;
