reserve X for set;
reserve CS for non empty CollStr;
reserve a,b,c for Point of CS;
reserve CLSP for CollSp;
reserve a,b,c,d,p,q,r for Point of CLSP;

theorem Th9:
  p<>q & a,b,p are_collinear & a,b,q are_collinear & p,q,r
  are_collinear implies a,b,r are_collinear
proof
  assume that
A1: p<>q and
A2: a,b,p are_collinear & a,b,q are_collinear and
A3: p,q,r are_collinear;
  now
    assume
A4: a<>b;
    then a,p,q are_collinear by A2,Th6;
    then
A5: p,q,a are_collinear by Th8;
    a,b,b are_collinear by Th2;
    then p,q,b are_collinear by A2,A4,Th3;
    hence thesis by A1,A3,A5,Th3;
  end;
  hence thesis by Th2;
end;
