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 Th34:
  X is being_plane implies ex a,b,c st a in X & b in X & c in X & not LIN a,b,c
proof
  assume X is being_plane;
  then consider K,P such that
A1: K is being_line and
A2: P is being_line and
A3: not K // P and
A4: X = Plane(K,P);
  consider a,b such that
A5: a in P and
A6: b in P and
A7: a<>b by A2,AFF_1:19;
  set Q = a*K;
  consider c such that
A8: a<>c and
A9: c in Q by A1,Th27,AFF_1:20;
  take a,b,c;
A10: P c= Plane(K,P) by A1,Th14;
  hence a in X & b in X by A4,A5,A6;
A11: K // Q & a in Q by A1,Def3;
  then Q c= Plane(K,P) by A5,A10,Lm4;
  hence c in X by A4,A9;
A12: Q is being_line by A1,Th27;
  thus not LIN a,b,c
  proof
    assume LIN a,b,c;
    then c in P by A2,A5,A6,A7,AFF_1:25;
    hence contradiction by A2,A3,A5,A11,A12,A8,A9,AFF_1:18;
  end;
end;
