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 Th28:
  (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)) implies
for p,p1,q,
  q1 ex r st p,p1,r are_collinear & q,q1,r are_collinear
proof
  given x1,x2 being Element of ProjectiveSpace(V) such that
A1: x1<>x2 and
A2: for r1,r2 ex q st x1,x2,q are_collinear & r1,r2,q are_collinear;
  let p,p1,q,q1;
  consider p3 being Element of ProjectiveSpace(V) such that
A3: x1,x2,p3 are_collinear and
A4: p,p1,p3 are_collinear by A2;
  consider q3 being Element of ProjectiveSpace(V) such that
A5: x1,x2,q3 are_collinear and
A6: q,q1,q3 are_collinear by A2;
  consider s2 being Element of ProjectiveSpace(V) such that
A7: x1,x2,s2 are_collinear and
A8: p,q,s2 are_collinear by A2;
A9: s2,p,q are_collinear by A8,Th24;
  consider s4 being Element of ProjectiveSpace(V) such that
A10: x1,x2,s4 are_collinear and
A11: p,q1,s4 are_collinear by A2;
A12: s4,q1,p are_collinear by A11,Th24;
  p3,s2,q3 are_collinear by A1,A3,A5,A7,Def8;
  then consider s3 being Element of ProjectiveSpace(V) such that
A13: p3,p,s3 are_collinear and
A14: q3,q,s3 are_collinear by A9,Def9;
  consider s being Element of ProjectiveSpace(V) such that
A15: x1,x2,s are_collinear and
A16: p1,q1,s are_collinear by A2;
  q3,s4,p3 are_collinear by A1,A3,A5,A10,Def8;
  then consider s5 being Element of ProjectiveSpace(V) such that
A17: q3,q1,s5 are_collinear and
A18: p3,p,s5 are_collinear by A12,Def9;
A19: p1,s,q1 are_collinear by A16,Th24;
  consider s6 being Element of ProjectiveSpace(V) such that
A20: x1,x2,s6 are_collinear and
A21: p1,q,s6 are_collinear by A2;
A22: s6,p1,q are_collinear by A21,Th24;
  p3,s6,q3 are_collinear by A1,A3,A5,A20,Def8;
  then consider s7 being Element of ProjectiveSpace(V) such that
A23: p3,p1,s7 are_collinear and
A24: q3,q,s7 are_collinear by A22,Def9;
  s,p3,q3 are_collinear by A1,A3,A5,A15,Def8;
  then consider s1 being Element of ProjectiveSpace(V) such that
A25: p1,p3,s1 are_collinear and
A26: q1,q3,s1 are_collinear by A19,Def9;
A27: now
A28: now
A29:  q3,q1,s1 are_collinear by A26,Th24;
      assume that
A30:  p3<>p1 and
A31:  q3<>q1;
      q3,q1,q are_collinear & q3,q1,q1 are_collinear by A6,Def7,Th24;
      then
A32:  q,q1,s1 are_collinear by A31,A29,Def8;
      take s1;
A33:  p3,p1,s1 are_collinear by A25,Th24;
      p3,p1,p are_collinear & p3,p1,p1 are_collinear by A4,Def7,Th24;
      then p,p1,s1 are_collinear by A30,A33,Def8;
      hence thesis by A32;
    end;
A34: now
      assume that
A35:  p3<>p and
A36:  q3<>q;
      take s3;
      q3,q,q are_collinear & q3,q,q1 are_collinear by A6,Def7,Th24;
      then
A37:  q,q1,s3 are_collinear by A14,A36,Def8;
      p3,p,p are_collinear & p3,p,p1 are_collinear by A4,Def7,Th24;
      then p,p1,s3 are_collinear by A13,A35,Def8;
      hence thesis by A37;
    end;
A38: now
      assume that
A39:  p3<>p1 and
A40:  q3<>q;
      take s7;
      q3,q,q are_collinear & q3,q,q1 are_collinear by A6,Def7,Th24;
      then
A41:  q,q1,s7 are_collinear by A24,A40,Def8;
      p3,p1,p are_collinear & p3,p1,p1 are_collinear by A4,Def7,Th24;
      then p,p1,s7 are_collinear by A23,A39,Def8;
      hence thesis by A41;
    end;
A42: now
      assume that
A43:  p3<>p and
A44:  q3<>q1;
      take s5;
      q3,q1,q are_collinear & q3,q1,q1 are_collinear by A6,Def7,Th24;
      then
A45:  q,q1,s5 are_collinear by A17,A44,Def8;
      p3,p,p are_collinear & p3,p,p1 are_collinear by A4,Def7,Th24;
      then p,p1,s5 are_collinear by A18,A43,Def8;
      hence thesis by A45;
    end;
    assume p<>p1 & q<>q1;
    hence thesis by A34,A42,A38,A28;
  end;
  now
A46: now
      assume
A47:  p=p1;
      take q3;
      p,p1,q3 are_collinear by A47,Def7;
      hence thesis by A6;
    end;
A48: now
      assume
A49:  q=q1;
      take p3;
      q,q1,p3 are_collinear by A49,Def7;
      hence thesis by A4;
    end;
    assume p = p1 or q = q1;
    hence thesis by A48,A46;
  end;
  hence thesis by A27;
end;
