reserve AS for AffinSpace;
reserve a,a9,b,b9,c,d,o,p,q,r,s,x,y,z,t,u,w for Element of AS;
reserve A,C,D,K for Subset of AS;

theorem
  x in K & not a in K & a,b // K implies (a=b or not LIN x,a,b)
proof
  assume that
A1: x in K and
A2: not a in K and
A3: a,b // K;
  set A=Line(a,b);
  assume that
A4: a<>b and
A5: LIN x,a,b;
  LIN a,b,x by A5,Th5;
  then
A6: x in A by Def2;
A7: a in A by Th14;
  A // K by A3,A4;
  hence contradiction by A1,A2,A6,A7,Th44;
end;
