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;
