theorem Th17:
  o<>b1 & a1<>b1 & o<>b2 & a2<>b2 & o<>b3 & a3<>b3 & not o,a1,a2
  are_collinear & not o,a1,a3 are_collinear & not o,a2,a3 are_collinear &
a1,a2,c3
  are_collinear & b1,b2,c3 are_collinear & a2,a3,c1 are_collinear & b2,b3,c1
  are_collinear & a1,a3,c2 are_collinear & b1,b3,c2 are_collinear & o,a1,b1
  are_collinear & o,a2,b2 are_collinear & o,a3,b3 are_collinear
implies c1,c2,c3
  are_collinear
proof
  assume that
A1: o<>b1 & a1<>b1 & o<>b2 & a2<>b2 & o<>b3 & a3<>b3 & ( not o,a1,a2
  are_collinear)& not o,a1,a3 are_collinear and
A2: not o,a2,a3 are_collinear and
A3: a1,a2,c3 are_collinear & b1,b2,c3 are_collinear and
A4: a2,a3,c1 are_collinear and
A5: b2,b3,c1 are_collinear and
A6: a1,a3,c2 are_collinear & b1,b3,c2 are_collinear & o,a1,b1
  are_collinear & o, a2,b2 are_collinear & o,a3,b3 are_collinear;
A7: o<>c1 by A2,A4,Th1;
A8: b3,b2,c1 are_collinear by A5,Th1;
A9: ( not o,a3,a2 are_collinear)& a3,a2,c1 are_collinear by A2,A4,Th1;
  now
    assume o,c1,c3 are_collinear;
    then
A10: c1,c3,o are_collinear by Th1;
    assume not thesis;
    then
A11: not c1,c3,c2 are_collinear by Th1;
    then not c1,o,c2 are_collinear by A7,A10,Th6;
    then not o,c1,c2 are_collinear by Th1;
    hence contradiction by A1,A3,A6,A9,A8,A11,Lm3;
  end;
  hence thesis by A1,A2,A3,A4,A5,A6,Lm3;
end;
