reserve AS for AffinSpace;
reserve a,b,c,d,a9,b9,c9,d9,p,q,r,x,y for Element of AS;
reserve A,C,K,M,N,P,Q,X,Y,Z for Subset of AS;

theorem Th19:
  X is being_plane & a in X & b in X & a<>b implies Line(a,b) c= X
proof
  assume that
A1: X is being_plane and
A2: a in X & b in X and
A3: a<>b;
  set Q = Line(a,b);
A4: a in Q & b in Q by AFF_1:15;
  Q is being_line & ex K,P st K is being_line & P is being_line & not K //
  P & X=Plane(K,P) by A1,A3,AFF_1:def 3;
  hence thesis by A2,A3,A4,Lm5;
end;
