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 Th26:
  p1,r2,q are_collinear & r1,q1,q are_collinear & p1,r1,p
  are_collinear & r2,q1,p are_collinear & p1,q1,r are_collinear & r2,r1,r
  are_collinear & p,q,r are_collinear implies
  (p1,r2,q1 are_collinear or p1,r2,r1
  are_collinear or p1,r1,q1 are_collinear or r2,r1,q1 are_collinear)
proof
  assume that
A1: p1,r2,q are_collinear and
A2: r1,q1,q are_collinear and
A3: p1,r1,p are_collinear and
A4: r2,q1,p are_collinear and
A5: p1,q1,r are_collinear and
A6: r2,r1,r are_collinear and
A7: p,q,r are_collinear;
  consider u1,w1,x being Element of V such that
A8: p1 = Dir(u1) and
A9: r1 = Dir(w1) and
A10: p = Dir(x) and
A11: u1 is not zero and
A12: w1 is not zero and
A13: x is not zero and
A14: u1,w1,x are_LinDep by A3,Th23;
  consider u,v,z being Element of V such that
A15: p1 = Dir(u) and
A16: r2 = Dir(v) and
A17: q = Dir(z) and
A18: u is not zero and
A19: v is not zero and
A20: z is not zero and
A21: u,v,z are_LinDep by A1,Th23;
  consider w,y,z1 being Element of V such that
A22: r1 = Dir(w) and
A23: q1 = Dir(y) and
A24: q = Dir(z1) and
A25: w is not zero and
A26: y is not zero and
A27: z1 is not zero and
A28: w,y,z1 are_LinDep by A2,Th23;
A29: are_Prop w1,w by A22,A25,A9,A12,ANPROJ_1:22;
  are_Prop z1,z by A17,A20,A24,A27,ANPROJ_1:22;
  then
A30: w,y,z are_LinDep by A28,ANPROJ_1:4;
  consider x2,z2,t2 being Element of V such that
A31: p = Dir(x2) and
A32: q = Dir(z2) and
A33: r = Dir(t2) and
A34: x2 is not zero and
A35: z2 is not zero and
A36: t2 is not zero and
A37: x2,z2,t2 are_LinDep by A7,Th23;
A38: are_Prop x2,x by A10,A13,A31,A34,ANPROJ_1:22;
  consider u2,y2,t being Element of V such that
A39: p1 = Dir(u2) and
A40: q1 = Dir(y2) and
A41: r = Dir(t) and
A42: u2 is not zero and
A43: y2 is not zero and
A44: t is not zero and
A45: u2,y2,t are_LinDep by A5,Th23;
A46: are_Prop y2,y by A23,A26,A40,A43,ANPROJ_1:22;
A47: are_Prop t2,t by A41,A44,A33,A36,ANPROJ_1:22;
  are_Prop z2,z by A17,A20,A32,A35,ANPROJ_1:22;
  then
A48: x,z,t are_LinDep by A37,A38,A47,ANPROJ_1:4;
  are_Prop u2,u by A15,A18,A39,A42,ANPROJ_1:22;
  then
A49: u,y,t are_LinDep by A45,A46,ANPROJ_1:4;
  consider v2,w2,t1 being Element of V such that
A50: r2 = Dir(v2) and
A51: r1 = Dir(w2) and
A52: r = Dir(t1) and
A53: v2 is not zero and
A54: w2 is not zero and
A55: t1 is not zero and
A56: v2,w2,t1 are_LinDep by A6,Th23;
A57: are_Prop t1,t by A41,A44,A52,A55,ANPROJ_1:22;
  are_Prop u1,u by A15,A18,A8,A11,ANPROJ_1:22;
  then
A58: u,w,x are_LinDep by A14,A29,ANPROJ_1:4;
  consider v1,y1,x1 being Element of V such that
A59: r2 = Dir(v1) and
A60: q1 = Dir(y1) and
A61: p = Dir(x1) and
A62: v1 is not zero and
A63: y1 is not zero and
A64: x1 is not zero and
A65: v1,y1,x1 are_LinDep by A4,Th23;
A66: are_Prop x1,x by A10,A13,A61,A64,ANPROJ_1:22;
A67: are_Prop w2,w by A22,A25,A51,A54,ANPROJ_1:22;
  are_Prop v2,v by A16,A19,A50,A53,ANPROJ_1:22;
  then
A68: v,w,t are_LinDep by A56,A67,A57,ANPROJ_1:4;
A69: are_Prop y1,y by A23,A26,A60,A63,ANPROJ_1:22;
  are_Prop v1,v by A16,A19,A59,A62,ANPROJ_1:22;
  then v,y,x are_LinDep by A65,A69,A66,ANPROJ_1:4;
  then u,v,y are_LinDep or u,v,w are_LinDep or u,w,y are_LinDep or v,w,y
  are_LinDep by A20,A21,A13,A44,A30,A58,A49,A68,A48,ANPROJ_1:18;
  hence thesis by A15,A16,A18,A19,A22,A23,A25,A26,Th23;
end;
