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 Th30:
  (ex y,u,v,w st (for w1 ex a,b,a1,b1 st w1 = a*y + b*u + a1*v +
b1*w) & (for a,b,a1,b1 st a*y + b*u + a1*v + b1*w = 0.V holds a=0 & b=0 & a1=0
& b1=0)) implies ex p,q1,q2 st not p,q1,q2 are_collinear & for r1,r2
 ex q3,r3 st
  r1,r2,r3 are_collinear & q1,q2,q3 are_collinear & p,r3,q3 are_collinear
proof
  given y,u,v,w such that
A1: for w1 ex a,b,a1,b1 st w1 = a*y + b*u + a1*v + b1*w and
A2: for a,b,a1,b1 st a*y + b*u + a1*v + b1*w = 0.V holds a=0 & b=0 & a1
  =0 & b1=0;
A3: u is not zero & v is not zero by A2,Th2;
A4: y is not zero by A2,Th2;
  then reconsider
  p = Dir(y),q1 = Dir(u),q2 = Dir(v) as Element of
  ProjectiveSpace(V) by A3,ANPROJ_1:26;
  take p,q1,q2;
  not y,u,v are_LinDep by A2,Th2;
  then not p,q1,q2 are_collinear by A4,A3,ANPROJ_1:25;
  hence not p,q1,q2 are_collinear;
  let r1,r2;
  consider u1 such that
A5: u1 is not zero and
A6: r1 = Dir(u1) by ANPROJ_1:26;
  consider u2 such that
A7: u2 is not zero and
A8: r2 = Dir(u2) by ANPROJ_1:26;
  consider w1,w2 such that
A9: u1,u2,w2 are_LinDep and
A10: u,v,w1 are_LinDep and
A11: y,w2,w1 are_LinDep and
A12: w1 is not zero and
A13: w2 is not zero by A1,A2,A5,A7,Th4;
  reconsider q3 = Dir(w1),r3 = Dir(w2) as Element of
  ProjectiveSpace(V) by A12,A13,ANPROJ_1:26;
  take q3,r3;
  thus r1,r2,r3 are_collinear by A5,A6,A7,A8,A9,A13,Th23;
  thus q1,q2,q3 are_collinear by A3,A10,A12,Th23;
  thus thesis by A4,A11,A12,A13,Th23;
end;
