reserve SAS for Semi_Affine_Space;
reserve a,a9,a1,a2,a3,a4,b,b9,c,c9,d,d9,d1,d2,o,p,p1,p2,q,r,r1,r2,s,x, y,t,z
  for Element of SAS;

theorem
  qtrap o,p implies ex q st not o,p,q are_collinear & qtrap o,q
proof
A1: o,p // o,p by Th1;
  assume
A2: qtrap o,p;
  then
A3: o<>p by Th103;
  consider r such that
A4: not o,p,r are_collinear by A2,Th25,Th103;
  consider q such that
A5: o,r,q are_collinear and
A6: qtrap o,q by Th101;
  take q;
  o<>q & o,r // o,q by A5,A6,Th103;
  hence thesis by A3,A4,A6,A1,Th23;
end;
