
theorem Th17:
  for R being Element of ProjectiveSpace TOP-REAL 3
  for P,Q being Element of BK_model
  for u,v,w being non zero Element of TOP-REAL 3
  for r being Real st 0 <= r <= 1 & P = Dir u &
  Q = Dir v & R = Dir w & u.3 = 1 & v.3 = 1 &
  w = r * u + (1 - r) * v holds R is Element of BK_model
  proof
    let R be Element of ProjectiveSpace TOP-REAL 3;
    let P,Q be Element of BK_model;
    let u,v,w be non zero Element of TOP-REAL 3;
    let r be Real;
    assume that
A1: 0 <= r <= 1 and
A2: P = Dir u & Q = Dir v & R = Dir w and
A3: u.3 = 1 & v.3 = 1 and
A4: w = r * u + (1 - r) * v;
    reconsider ru = r * u,rv  = (1 - r) * v as Element of REAL 3 by EUCLID:22;
A5: w.3 = ru.3 + rv.3 by A4,RVSUM_1:11
       .= r * u.3 + rv.3 by RVSUM_1:44
       .= r + (1 - r) * 1 by A3,RVSUM_1:44
       .= 1;
    consider u2 be non zero Element of TOP-REAL 3 such that
A6: Dir u2 = P & u2.3 = 1 & BK_to_REAL2 P = |[u2.1,u2.2]| by BKMODEL2:def 2;
A7: u = u2 by A2,A3,A6,Th16;
    reconsider ru2 = |[u2.1,u2.2]| as Element of TOP-REAL 2;
    consider v2 be non zero Element of TOP-REAL 3 such that
A8: Dir v2 = Q & v2.3 = 1 & BK_to_REAL2 Q = |[v2.1,v2.2]| by BKMODEL2:def 2;
A9: v = v2 by A2,A3,A8,Th16;
    reconsider rv2 = |[v2.1,v2.2]| as Element of TOP-REAL 2;
    reconsider rw = |[w.1,w.2]| as Element of TOP-REAL 2;
    rw = r * ru2 + (1 - r) * rv2
    proof
A10:  w.1 = ru.1 + rv.1 by A4,RVSUM_1:11
                .= r * u.1 + rv.1 by RVSUM_1:44
                .= r * u2.1 + (1 - r) * v2.1 by A7,A9,RVSUM_1:44;
A13:  w.2 = ru.2 + rv.2 by A4,RVSUM_1:11
                .= r * u.2 + rv.2 by RVSUM_1:44
                .= r * u2.2 + (1 - r) * v2.2 by A7,A9,RVSUM_1:44;
      r * ru2 + (1 - r) * rv2 = |[r*u2.1,r*u2.2]| + (1 - r) * |[v2.1,v2.2]|
                                 by EUCLID:58
                             .= |[r*u2.1,r*u2.2]| + |[(1-r)*v2.1,(1-r)*v2.2]|
                                 by EUCLID:58;
      hence thesis by A10,A13,EUCLID:56;
    end;
    then rw = (1 - r) * rv2 + r * ru2;
    then rw in {(1 - r) * rv2 + r * ru2 where r is Real:0 <= r & r <= 1} by A1;
    then rw in LSeg(rv2,ru2) by RLTOPSP1:def 2;
    then reconsider rw as Element of inside_of_circle(0,0,1) by A6,A8,Th15;
    consider RW2 be Element of TOP-REAL 2 such that
A11: RW2 = rw & REAL2_to_BK rw = Dir |[RW2`1,RW2`2,1]| by BKMODEL2:def 3;
A12: rw`1 = w.1 & rw`2 = w.2 by EUCLID:52;
    w = |[w`1,w`2,w`3]| by EUCLID_5:3
     .= |[w.1,w`2,w`3]| by EUCLID_5:def 1
     .= |[w.1,w.2,w`3]| by EUCLID_5:def 2
     .= |[w.1,w.2,w.3]| by EUCLID_5:def 3;
    hence thesis by A2,A11,A5,A12;
  end;
