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;
reserve AP for AffinPlane;
reserve a,b,c,d,x,p,q for Element of AP;
reserve A,C for Subset of AP;

theorem
  A is being_line & not a,b // A implies ex x st x in A & LIN a,b,x
proof
  assume that
A1: A is being_line and
A2: not a,b // A;
  set C=Line(a,b);
A3: not C // A
  proof
A4: b in C by Th14;
    assume C // A;
    then consider p,q such that
A5: C=Line(p,q) and
A6: p<>q and
A7: p,q // A;
    a in C by Th14;
    then p,q // a,b by A5,A6,A4,Th21;
    hence contradiction by A2,A6,A7,Th31;
  end;
  a<>b by A1,A2,Th32;
  then C is being_line;
  then consider x such that
A8: x in C and
A9: x in A by A1,A3,Th57;
  LIN a,b,x by A8,Def2;
  hence thesis by A9;
end;
