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 Th101:
  for o,a holds ex p st o,a,p are_collinear & qtrap o,p
proof
  let o,a;
  consider p such that
A1: for b,c holds o,a // o,p & ex d st o,p // o,b implies o,c // o,d & p
  ,c // b,d by Def1;
  take p;
  now
    thus o,a,p are_collinear by A1;
    let b,c;
    consider d such that
A2: o,p // o,b implies o,c // o,d & p,c // b,d by A1;
    take d;
    assume o,p,b are_collinear;
    hence o,c,d are_collinear & p,c // b,d by A2;
  end;
  hence thesis;
end;
