
theorem
  for X be RealNormSpace st X is complete
  holds product <*X*> is complete
  proof
    let X be RealNormSpace;
    A1: dom <*X*> = {1} by FINSEQ_1:2,38; then
    reconsider i = 1 as Element of dom <*X*> by TARSKI:def 1;
    assume
    A2: X is complete;

    now
      let i be Element of dom <*X*>;
      i = 1 by A1,TARSKI:def 1;
      hence <*X*>.i is complete by A2;
    end;
    hence product <*X*> is complete by PRVECT_2:14;
  end;
