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 Th27:
  (ex u,v,w st (for a,b,c st a*u + b*v + c*w = 0.V holds a=0 & b=0
  & c = 0) & (for y ex a,b,c st y = a*u + b*v + c*w)) implies ex x1,x2 being
  Element of ProjectiveSpace(V) st (x1<>x2 & for r1,r2 ex q st x1,x2,q
  are_collinear & r1,r2,q are_collinear)
proof
  given p,q,r being Element of V such that
A1: for a,b,c st a*p + b*q + c*r = 0.V holds a=0 & b=0 & c = 0 and
A2: for y ex a,b,c st y = a*p + b*q + c*r;
A3: p is not zero & q is not zero by A1,Th1;
  then reconsider x1=Dir(p),x2=Dir(q) as Element of ProjectiveSpace(V) by
ANPROJ_1:26;
  take x1,x2;
  not are_Prop p,q by A1,Th1;
  hence x1<>x2 by A3,ANPROJ_1:22;
  let r1,r2;
  consider u such that
A4: u is not zero & r1 = Dir(u) by ANPROJ_1:26;
  consider u1 such that
A5: u1 is not zero & r2 = Dir(u1) by ANPROJ_1:26;
  consider y such that
A6: p,q,y are_LinDep and
A7: u,u1,y are_LinDep and
A8: y is not zero by A1,A2,Th3;
  reconsider q = Dir(y) as Element of ProjectiveSpace(V) by A8,ANPROJ_1:26;
  take q;
  thus x1,x2,q are_collinear by A3,A6,A8,Th23;
  thus thesis by A4,A5,A7,A8,Th23;
end;
