reserve V for RealLinearSpace,
  o,p,q,r,s,u,v,w,y,y1,u1,v1,w1,u2,v2,w2 for Element of V,
  a,b,c,d,a1,b1,c1,d1,a2,b2,c2,d2,a3,b3,c3,d3 for Real,
  z for set;
reserve A for non empty set;
reserve f,g,h,f1 for Element of Funcs(A,REAL);
reserve x1,x2,x3,x4 for Element of A;
reserve V for non trivial RealLinearSpace;
reserve u,v,w,y,u1,v1,w1,u2,w2 for Element of V;
reserve p,p1,p2,p3,q,q1,q2,q3,r,r1,r2,r3 for Element of ProjectiveSpace(V);

theorem Th24:
  p,q,r are_collinear implies p,r,q are_collinear & q,p,r
  are_collinear & r,q,p are_collinear & r,p,q are_collinear &
  q,r,p are_collinear
proof
  assume p,q,r are_collinear;
  then consider u,v,w such that
A1: p = Dir(u) & q = Dir(v) & r = Dir(w) & u is not zero & v is not zero
  & w is not zero and
A2: u,v,w are_LinDep by Th23;
A3: w,v,u are_LinDep & w,u,v are_LinDep by A2,ANPROJ_1:5;
A4: v,w,u are_LinDep by A2,ANPROJ_1:5;
  u,w,v are_LinDep & v,u,w are_LinDep by A2,ANPROJ_1:5;
  hence thesis by A1,A3,A4,Th23;
end;
