theorem
  A1 <> {} & A1 c= A2 implies for x st x is Vector of A1 holds x is Vector of
  A2
proof
  assume that
A1: A1 <> {} and
A2: A1 c= A2;
  let x;
  assume x is Vector of A1;
  then reconsider a=x as Vector of A1;
  a is Element of A1 by A1,Def11;
  then a in A2 by A1,A2,TARSKI:def 3;
  hence thesis by Def11;
end;
