
theorem RHS11b:
  for X be RealUnitarySpace holds
    X is complete iff RUSp2RNSp X is complete
proof
  let X be RealUnitarySpace;
  set Y = RUSp2RNSp X;
  hereby assume AS1: X is complete;
   for s2 be sequence of Y holds
     s2 is Cauchy_sequence_by_Norm implies s2 is convergent
   proof
     let s2 be sequence of Y;
     reconsider s1=s2 as sequence of X;
     assume s2 is Cauchy_sequence_by_Norm; then
     s1 is Cauchy by RHS11a; then
     s1 is convergent by AS1,BHSP_3:def 4;
     hence s2 is convergent by RHS8;
   end;
   hence Y is complete by LOPBAN_1:def 15;
  end;
  assume AS2: Y is complete;
  for s1 be sequence of X holds
     s1 is Cauchy implies s1 is convergent
  proof
    let s1 be sequence of X;
    reconsider s2=s1 as sequence of Y;
    assume s1 is Cauchy; then
    s2 is Cauchy_sequence_by_Norm by RHS11a; then
    s2 is convergent by AS2,LOPBAN_1:def 15;
    hence s1 is convergent by RHS8;
  end;
  hence X is complete by BHSP_3:def 4;
end;
