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
  A is being_line implies ( a in A iff a*A = A )
proof
  assume
A1: A is being_line;
  now
    assume
A2: a in A;
    A // A by A1,AFF_1:41;
    hence a*A = A by A1,A2,Def3;
  end;
  hence thesis by A1,Def3;
end;
