
theorem RNSBH1:
  for X be RealNormSpace, x be Point of X,
    vseq be sequence of DualSp X,
    vseq1 be sequence of R_NormSpace_of_BoundedLinearOperators(X,RNS_Real)
  st vseq = vseq1 holds vseq#x = vseq1#x
proof
  let X be RealNormSpace, x be Point of X,
      vseq be sequence of DualSp X,
      vseq1 be sequence of R_NormSpace_of_BoundedLinearOperators(X,RNS_Real);
  assume AS: vseq = vseq1;
  for n be Element of NAT holds (vseq#x).n = (vseq1#x).n
  proof
    let n be Element of NAT;
    (vseq#x).n = (vseq.n).x by Def1;
    hence (vseq#x).n = (vseq1#x).n by AS,LOPBAN_5:def 2;
  end;
  hence thesis;
end;
