
theorem Th18:
  for P1,P2 being Element of absolute
  for Q being Element of BK_model
  for u,v,w being non zero Element of TOP-REAL 3 st
  Dir u = P1 & Dir v = P2 & Dir w = Q &
  u.3 = 1 & v.3 = 1 & w.3 = 1 & v.1 = - u.1 & v.2 = - u.2 &
  P1,Q,P2 are_collinear holds
  ex a being Real st -1 < a < 1 & w.1 = a * u.1 & w.2 = a * u.2
  proof
    let P1,P2 be Element of absolute;
    let Q be Element of BK_model;
    let u,v,w be non zero Element of TOP-REAL 3;
    assume that
A1: Dir u = P1 & Dir v = P2 & Dir w = Q and
A2: u.3 = 1 & v.3 = 1 & w.3 = 1 and
A3: v.1 = - u.1 & v.2 = - u.2 and
A4: P1,Q,P2 are_collinear;
    u.1 = u`1 & u.2 = u`2 by EUCLID_5:def 1,def 2;
    then
A6: u`3 = 1 & v`3 = 1 & w`3 = 1 & v`1 = - u`1 & v`2 = - u`2
      by A2,A3,EUCLID_5:def 1,def 2,def 3;
    P1,P2,Q are_collinear by A4,COLLSP:4; then
A7: 0 = |{ u,v,w }| by A1,BKMODEL1:1
     .= u`1 * (-u`2) * 1 - 1 * (-u`2) * w`1 - u`1 * 1 * w`2
        + u`2 * 1 * w`1 - u`2 * (-u`1) * 1 + 1 * (-u`1) * w`2 by A6,ANPROJ_8:27
     .= 2 * (u`2 * w`1 - u`1 * w`2);
    consider u9 be non zero Element of TOP-REAL 3 such that
A8: (u9.1)^2 + (u9.2)^2 = 1 and
A9: u9.3 = 1 and
A10: P1 = Dir u9 by BKMODEL1:89;
A11: u = u9 by A9,A10,A1,A2,BKMODEL1:43;
    not (u`1 = 0 & u`2 = 0)
    proof
      assume u`1 = 0 & u`2 = 0;
      then u.1 = 0 & u.2 = 0 by EUCLID_5:def 1,def 2;
      hence contradiction by A11,A8;
    end;
    then consider e be Real such that
A13: w`1 = e * u`1 & w`2 = e * u`2 by A7,BKMODEL1:2;
    w.1 = e * u`1 & w.2 = e * u`2 by A13,EUCLID_5:def 1,def 2;
    then
A14: w.1 = e * u.1 & w.2 = e * u.2 by EUCLID_5:def 1,def 2;
    per cases;
    suppose e = 0;
      hence thesis by A14;
    end;
    suppose e <> 0;
      (w.1)^2 + (w.2)^2 = w`1 * w.1 + w.2 * w.2 by EUCLID_5:def 1
                       .= w`1 * w`1 + w.2 * w.2 by EUCLID_5:def 1
                       .= w`1 * w`1 + w`2 * w.2 by EUCLID_5:def 2
                       .= w`1 * w`1 + w`2 * w`2 by EUCLID_5:def 2
                       .= e * e * (u`1 * u`1 + u`2 * u`2) by A13
                       .= e * e * (u.1 * u`1 + u`2 * u`2) by EUCLID_5:def 1
                       .= e * e * (u.1 * u.1 + u`2 * u`2) by EUCLID_5:def 1
                       .= e * e * (u.1 * u.1 + u.2 * u`2) by EUCLID_5:def 2
                       .= e * e * (u.1 * u.1 + u.2 * u.2) by EUCLID_5:def 2
                       .= e * e by A8,A11;
      then e^2 < 1 by A1,A2,Th17;
      then - 1 < e < 1 by SQUARE_1:52;
      hence thesis by A14;
    end;
  end;
