reserve PCPP for CollProjectiveSpace,
  a,a9,a1,a2,a3,b,b9,b1,b2,c,c1,c3,d,d9,e,
  o,p,p1,p2,p3,r,q, q1,q2,q3,x,y for Element of PCPP;
reserve PCPP for Pappian CollProjectivePlane,
  a,a1,a2,a3,b1,b2,b3,c1,c2,c3,o,p
  ,p1,p2,p3,q,q9, q1,q2,q3,r,r1,r2,r3,x,y,z for Element of PCPP;

theorem Th16:
  p2<>p3 & p1<>p3 & q2<>q3 & q1<>q2 & q1<>q3 & not p1,p2,q1
  are_collinear & p1,p2,p3 are_collinear & q1,q2,q3 are_collinear & p1,q2,r3
  are_collinear & q1,p2,r3 are_collinear & p1,q3,r2 are_collinear & p3,q1,r2
  are_collinear & p2,q3,r1 are_collinear & p3,q2,r1 are_collinear
implies r1,r2,r3
  are_collinear
proof
  assume that
A1: p2<>p3 and
A2: p1<>p3 and
A3: q2<>q3 and
A4: q1<>q2 and
A5: q1<>q3 and
A6: not p1,p2,q1 are_collinear and
A7: p1,p2,p3 are_collinear and
A8: q1,q2,q3 are_collinear and
A9: p1,q2,r3 are_collinear and
A10: q1,p2,r3 are_collinear and
A11: p1,q3,r2 are_collinear and
A12: p3,q1,r2 are_collinear and
A13: p2,q3,r1 are_collinear and
A14: p3,q2,r1 are_collinear;
A15: p1<>p2 by A6,ANPROJ_2:def 7;
A16: now
    assume
A17: not p1,p2,q2 are_collinear;
A18: now
      assume
A19:  not p1,p2,q3 are_collinear;
A20:  now
A21:    now
          ( not p2,p1,q2 are_collinear)& p2,p1,p3 are_collinear by A7,A17,Th1;
          then not p2,p3,q2 are_collinear by A1,Th6;
          then
A22:      not q2,p2,p3 are_collinear by Th1;
          assume
A23:      q1,q2,p2 are_collinear;
          then
A24:      p2,q1,q2 are_collinear by Th1;
          p2<>q1 & p2,q1,r3 are_collinear by A6,A10,Th1,ANPROJ_2:def 7;
          then p2,r3,q2 are_collinear by A24,Th2;
          then
A25:      q2,p2,r3 are_collinear by Th1;
          q2,p1,r3 are_collinear & not q2,p1,p2 are_collinear by A9,A17,Th1;
          then
A26:      q2=r3 by A25,Th7;
A27:      q2,q1,q3 are_collinear by A8,Th1;
          q2,q1,p2 are_collinear by A23,Th1;
          then q2,p2,q3 are_collinear by A4,A27,Th2;
          then
A28:      p2,q3,q2 are_collinear by Th1;
          p2<>q3 by A19,ANPROJ_2:def 7;
          then p2,r1,q2 are_collinear by A13,A28,Th2;
          then
A29:      q2,p2,r1 are_collinear by Th1;
          q2,p3,r1 are_collinear by A14,Th1;
          then q2=r1 by A29,A22,Th7;
          hence thesis by A26,ANPROJ_2:def 7;
        end;
A30:    now
          ( not p2,p1,q3 are_collinear)& p2,p1,p3 are_collinear by A7,A19,Th1;
          then not p2,p3,q3 are_collinear by A1,Th6;
          then
A31:      not q3,p2,p3 are_collinear by Th1;
          assume
A32:      q1,q2,p3 are_collinear;
          then
A33:      q2,q1,p3 are_collinear by Th1;
          q2,q1,q3 are_collinear by A8,Th1;
          then q2,q3, p3 are_collinear by A4,A33,Th2;
          then p3,q2,q3 are_collinear by Th1;
          then p3,r1,q3 are_collinear by A7,A14,A17,Th2;
          then
A34:      q3,p3,r1 are_collinear by Th1;
          not p1,p3,q3 are_collinear by A2,A7,A19,Th6;
          then
A35:      not q3,p1,p3 are_collinear by Th1;
          q1,q3,p3 are_collinear by A4,A8,A32,Th2;
          then p3,q1,q3 are_collinear by Th1;
          then p3,r2,q3 are_collinear by A6,A7,A12,Th2;
          then
A36:      q3,p3,r2 are_collinear by Th1;
          q3,p2,r1 are_collinear by A13,Th1;
          then
A37:      q3=r1 by A34,A31,Th7;
          q3,p1,r2 are_collinear by A11,Th1;
          then q3=r2 by A36,A35,Th7;
          hence thesis by A37,ANPROJ_2:def 7;
        end;
A38:    now
          not p1,p3,q1 are_collinear by A2,A6,A7,Th6;
          then
A39:      not q1,p1,p3 are_collinear by Th1;
          assume
A40:      q1,q2,p1 are_collinear;
          then q1,p1,q3 are_collinear by A4,A8,Th2;
          then
A41:      p1,q3,q1 are_collinear by Th1;
          p1<>q3 by A19,ANPROJ_2:def 7;
          then p1,r2,q1 are_collinear by A11,A41,Th2;
          then
A42:      q1,p1,r2 are_collinear by Th1;
A43:      p1<>q2 by A17,ANPROJ_2:def 7;
          p1,q2,q1 are_collinear by A40,Th1;
          then p1,r3,q1 are_collinear by A9,A43,Th2;
          then
A44:      q1,p1,r3 are_collinear by Th1;
          not q1,p2,p1 are_collinear by A6,Th1;
          then
A45:      q1=r3 by A10,A44,Th7;
          q1,p3,r2 are_collinear by A12,Th1;
          then q1=r2 by A42,A39,Th7;
          hence thesis by A45,ANPROJ_2:def 7;
        end;
        assume q1,q2,p1 are_collinear or q1,q2,p2 are_collinear or q1,q2,p3
        are_collinear;
        hence thesis by A38,A21,A30;
      end;
      now
        assume that
A46:    not q1,q2,p1 are_collinear and
A47:    ( not q1,q2,p2 are_collinear)& not q1,q2,p3 are_collinear;
        consider o such that
A48:    p1,p2,o are_collinear and
A49:    q1,q2,o are_collinear by ANPROJ_2:def 14;
        not p1,o,q1 are_collinear by A6,A46,A48,A49,Th6;
        then
A50:    not o,p1,q1 are_collinear by Th1;
        q1,q3,o are_collinear by A4,A8,A49,Th2;
        then
A51:    o,q1,q3 are_collinear by Th1;
        p1,p3,o are_collinear by A7,A15,A48,Th2;
        then
A52:    o,p1,p3 are_collinear by Th1;
        o,p1,p2 are_collinear & o,q1,q2 are_collinear by A48,A49,Th1;
        hence
        thesis by A1,A2,A3,A4,A5,A9,A10,A11,A12,A13,A14,A15,A17,A19,A47,A48,A49
,A52,A51,A50,ANPROJ_2:def 13;
      end;
      hence thesis by A20;
    end;
    now
      assume
A53:  p1,p2,q3 are_collinear;
A54:  now
        assume that
A55:    p2<>q3 and
A56:    p1<>q3;
        p1,q3,p2 are_collinear by A53,Th1;
        then p1,p2,r2 are_collinear by A11,A56,Th2;
        then p1,r2,p3 are_collinear by A7,A15,Th2;
        then
A57:    p3,p1,r2 are_collinear by Th1;
        not p1,p3,q1 are_collinear by A2,A6,A7,Th6;
        then not p3,q1,p1 are_collinear by Th1;
        then
A58:    p3=r2 by A12,A57,Th7;
A59:    p2,p1,p3 are_collinear by A7,Th1;
        p2,q3,p1 are_collinear by A53,Th1;
        then p2,p1,r1 are_collinear by A13,A55,Th2;
        then p2,r1,p3 are_collinear by A15,A59,Th2;
        then
A60:    p3,p2,r1 are_collinear by Th1;
        ( not p2,p1,q2 are_collinear)& p2,p1,p3 are_collinear by A7,A17,Th1;
        then not p2,p3,q2 are_collinear by A1,Th6;
        then not p3,q2,p2 are_collinear by Th1;
        then p3=r1 by A14,A60,Th7;
        hence thesis by A58,ANPROJ_2:def 7;
      end;
A61:  now
        not p1,p3,q1 are_collinear by A2,A6,A7,Th6;
        then
A62:    not p3,p1,q1 are_collinear by Th1;
A63:    not p2,q1,p1 are_collinear by A6,Th1;
A64:    p2,q1,r3 are_collinear by A10,Th1;
        assume
A65:    p2=q3;
        then p2,q1,q2 are_collinear by A8,Th1;
        then p2,q2,r3 are_collinear by A5,A65,A64,Th2;
        then
A66:    q2,p2,r3 are_collinear by Th1;
        p2,q1,q2 are_collinear by A8,A65,Th1;
        then not p2,q2,p1 are_collinear by A3,A65,A63,Th6;
        then
A67:    not q2,p1,p2 are_collinear by Th1;
        p1,p3,r2 are_collinear by A7,A11,A15,A65,Th2;
        then p3,p1,r2 are_collinear by Th1;
        then
A68:    p3=r2 by A12,A62,Th7;
        q2,p1,r3 are_collinear by A9,Th1;
        then q2=r3 by A66,A67,Th7;
        hence thesis by A14,A68,Th1;
      end;
      now
        assume
A69:    p1=q3;
        then p1,p2,r1 are_collinear by A13,Th1;
        then p1,p3,r1 are_collinear by A7,A15,Th2;
        then
A70:    p3,p1,r1 are_collinear by Th1;
        p1,q2,q1 are_collinear by A8,A69,Th1;
        then p1,q1,r3 are_collinear by A3,A9,A69,Th2;
        then
A71:    q1,p1,r3 are_collinear by Th1;
        not p3,p1,q2 are_collinear by A2,A3,A6,A7,A8,A69,Th13;
        then
A72:    p3=r1 by A14,A70,Th7;
        not q1,p1,p2 are_collinear by A6,Th1;
        then q1=r3 by A10,A71,Th7;
        hence thesis by A12,A72,Th1;
      end;
      hence thesis by A61,A54;
    end;
    hence thesis by A18;
  end;
  now
A73: now
      not p1,p3,q1 are_collinear by A2,A6,A7,Th6;
      then
A74:  not q1,p1,p3 are_collinear by Th1;
A75:  not p1,q1,p2 are_collinear by A6,Th1;
      assume
A76:  p1=q2;
      then p1,q3,q1 are_collinear by A8,Th1;
      then p1,q1,r2 are_collinear by A3,A11,A76,Th2;
      then
A77:  q1,p1,r2 are_collinear by Th1;
A78:  p1,p3,p2 are_collinear by A7,Th1;
      p1,p3,r1 are_collinear by A14,A76,Th1;
      then p1,p2,r1 are_collinear by A2,A78,Th2;
      then
A79:  p2,p1,r1 are_collinear by Th1;
      q1,p3,r2 are_collinear by A12,Th1;
      then
A80:  q1=r2 by A77,A74,Th7;
      p1,q1,q3 are_collinear by A8,A76,Th1;
      then not p1,q3,p2 are_collinear by A3,A76,A75,Th6;
      then not p2,p1,q3 are_collinear by Th1;
      then r1=p2 by A13,A79,Th7;
      hence thesis by A10,A80,Th1;
    end;
    assume
A81: p1,p2,q2 are_collinear;
A82: now
      assume that
A83:  p1<>q2 and
A84:  p3<>q2;
      p1,q2,p2 are_collinear by A81,Th1;
      then p1,p2,r3 are_collinear by A9,A83,Th2;
      then
A85:  p2,p1,r3 are_collinear by Th1;
      p1,q2,p3 are_collinear by A7,A15,A81,Th2;
      then p3,q2,p1 are_collinear by Th1;
      then p3,p1,r1 are_collinear by A14,A84,Th2;
      then
A86:  p1,p3,r1 are_collinear by Th1;
      p1,p3,p2 are_collinear by A7,Th1;
      then p1,p2,r1 are_collinear by A2,A86,Th2;
      then
A87:  p2,p1,r1 are_collinear by Th1;
      not p2,p1,q3 are_collinear by A3,A6,A8,A81,Th11;
      then
A88:  p2=r1 by A13,A87,Th7;
      ( not p2,p1,q1 are_collinear)& p2,q1,r3 are_collinear by A6,A10,Th1;
      then p2=r3 by A85,Th7;
      hence thesis by A88,ANPROJ_2:def 7;
    end;
    now
      assume
A89:  p3=q2;
      then q1,q3,p3 are_collinear by A8,Th1;
      then
A90:  not q3,p1,q1 are_collinear by A2,A5,A6,A7,Th10;
      p1,p3,p2 are_collinear by A7,Th1;
      then p1,p2,r3 are_collinear by A2,A9,A89,Th2;
      then
A91:  p2,p1,r3 are_collinear by Th1;
      ( not p2,p1,q1 are_collinear)& p2,q1,r3 are_collinear by A6,A10,Th1;
      then
A92:  p2=r3 by A91,Th7;
      q1,q2,r2 are_collinear by A12,A89,Th1;
      then q1,q3,r2 are_collinear by A4,A8,Th2;
      then
A93:  q3,q1,r2 are_collinear by Th1;
      q3,p1,r2 are_collinear by A11,Th1;
      then r3,r2,r1 are_collinear by A13,A92,A90,A93,Th7;
      hence thesis by Th1;
    end;
    hence thesis by A73,A82;
  end;
  hence thesis by A16;
end;
