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
  X is being_plane implies ( a in X iff a+X = X )
proof
  assume
A1: X is being_plane;
  now
    assume
A2: a in X;
    X '||' X by A1,Th57;
    hence a+X = X by A1,A2,Def6;
  end;
  hence thesis by A1,Def6;
end;
