reserve a,b,c,d,e,f,g,h,i for Real,
                        M for Matrix of 3,REAL;

theorem Th2:
  for P1,P4,P5 being Element of ProjectiveSpace TOP-REAL 3
  for p1,p2,p3,p4,p5 being Element of TOP-REAL 3 st
  p1 is non zero & P1 = Dir p1 &
  p4 is non zero & P4 = Dir p4 &
  p5 is non zero & P5 = Dir p5 &
  P1,P4,P5 are_collinear
  holds
  |{p1,p2,p4}| * |{p1,p3,p5}| = |{p1,p2,p5}| * |{p1,p3,p4}|
  proof
    let P1,P4,P5 being Element of ProjectiveSpace TOP-REAL 3;
    let p1,p2,p3,p4,p5 being Element of TOP-REAL 3;
    assume that
A1: p1 is non zero & P1 = Dir p1 and
A2: p4 is non zero & P4 = Dir p4 and
A3: p5 is non zero & P5 = Dir p5 and
A4: P1,P4,P5 are_collinear;
A5: |{p1,p2,p3}| * |{p1,p4,p5}| - |{p1,p2,p4}| * |{p1,p3,p5}|
      + |{p1,p2,p5}| * |{p1,p3,p4}| = 0 by ANPROJ_8:28;
    |{p1,p4,p5}| = 0 by A1,A2,A3,A4,BKMODEL1:1;
    hence thesis by A5;
  end;
