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;
