
theorem
  for X be RealNormSpace st X is non trivial Reflexive holds
    X is RealBanachSpace
proof
   let X be RealNormSpace;
   assume AS: X is non trivial Reflexive; then
P1:BidualFunc X is onto;
  for seq be sequence of X
    st seq is Cauchy_sequence_by_Norm holds seq is convergent
  proof
   let seq be sequence of X;
   assume P2: seq is Cauchy_sequence_by_Norm;
   reconsider seq1= (BidualFunc X) * seq as sequence of DualSp DualSp X;
XX:BidualFunc X is additive homogeneous;
   for r be Real st r > 0
    ex k be Nat st for n, m be Nat st n >= k & m >= k
     holds ||.(seq1.n) - (seq1.m).|| < r
   proof
    let r be Real;
    assume r > 0; then
    consider k be Nat such that
A1:  for n, m be Nat st n >= k & m >= k
        holds ||.(seq.n) - (seq.m).|| < r by P2,RSSPACE3:8;
AK: for n, m be Nat st n >= k & m >= k holds ||.(seq1.n) - (seq1.m).|| < r
    proof
     let n, m be Nat;
     assume n >= k & m >= k; then
A2:  ||.(seq.n) - (seq.m).|| < r by A1;
     n in NAT & m in NAT by ORDINAL1:def 12; then
     n in dom seq & m in dom seq by FUNCT_2:def 1; then
A4:  seq1.n = (BidualFunc X).(seq.n)
   & seq1.m = (BidualFunc X).(seq.m) by FUNCT_1:13;
     seq.n - seq.m = seq.n + (-1)*seq.m by RLVECT_1:16; then
A7:  (BidualFunc X).(seq.n - seq.m)
        = (BidualFunc X).(seq.n) + (BidualFunc X).((-1)*seq.m) by XX;
     (BidualFunc X).((-1)*seq.m)
        = (-1)*(BidualFunc X).(seq.m) by LOPBAN_1:def 5; then
     (BidualFunc X).(seq.n - seq.m)
        = (BidualFunc X).(seq.n) - (BidualFunc X).(seq.m) by A7,RLVECT_1:16;
     hence thesis by A2,AS,A4,LMNORM;
    end;
    take k;
    thus thesis by AK;
   end; then
P5:seq1 is convergent by LOPBAN_1:def 15,RSSPACE3:8;
   consider s be Point of X such that
P7: lim seq1 = (BidualFunc X).s by P1,FUNCT_2:113;
   for r be Real st 0 < r ex m be Nat st for n be Nat
     st m <= n holds ||.(seq.n) - s .|| < r
   proof
    let r be Real;
    assume 0 < r; then
    consider m be Nat such that
B1:   for n be Nat st m <= n holds ||.(seq1.n) - lim seq1 .|| < r
         by P5,NORMSP_1:def 7;
BK: for n be Nat st m <= n holds ||.(seq.n) - s .|| < r
    proof
     let n be Nat;
     assume m <= n; then
B2:  ||.(seq1.n) - lim seq1 .|| < r by B1;
     n in NAT by ORDINAL1:def 12; then
     n in dom seq by FUNCT_2:def 1; then
B4:  seq1.n = (BidualFunc X).(seq.n) by FUNCT_1:13;
     seq.n - s = seq.n + (-1)*s by RLVECT_1:16; then
B6:  (BidualFunc X).(seq.n - s)
        = (BidualFunc X).(seq.n) + (BidualFunc X).((-1)*s) by XX;
     (BidualFunc X).((-1)*s)
        = (-1)*(BidualFunc X).s by LOPBAN_1:def 5; then
     (BidualFunc X).(seq.n - s)
        = (BidualFunc X).(seq.n) - (BidualFunc X).s by B6,RLVECT_1:16;
     hence thesis by B2,AS,P7,B4,LMNORM;
    end;
    take m;
    thus thesis by BK;
   end;
   hence seq is convergent;
  end;
  hence thesis by LOPBAN_1:def 15;
end;
