
theorem
  for X be RealBanachSpace, X0 be non empty Subset of X
    st X is non trivial Reflexive holds
      X0 is weakly-sequentially-compact iff
      ex S be non empty Subset of REAL st
         S = {||.x.|| where x is Point of X : x in X0}
       & S is bounded_above
proof
  let X be RealBanachSpace, X0 be non empty Subset of X;
  assume AS: X is non trivial Reflexive;
  hence X0 is weakly-sequentially-compact implies
    ex S be non empty Subset of REAL st
       S = {||.x.|| where x is Point of X : x in X0}
     & S is bounded_above by Th713A;
    given S be non empty Subset of REAL such that
B0:   S = {||.x.|| where x is Point of X : x in X0}
    & S is bounded_above;
    for seq be sequence of X0
      ex seq1 be sequence of X st
        seq1 is subsequence of seq & seq1 is weakly-convergent
      & w-lim seq1 in X
    proof
      let seq0 be sequence of X0;
      reconsider seq=seq0 as sequence of X by FUNCT_2:7;
      consider r be Real such that
B1:     r is UpperBound of S by B0;
B2:   r + 0 < r + 1 by XREAL_1:8;
      for n be Nat holds ||.seq.||.n < (r+1)
      proof
        let n be Nat;
        seq0.n in X0; then
        ||.seq.n.|| in S by B0; then
        ||.seq.n.|| <= r by B1,XXREAL_2:def 1; then
        ||.seq.||.n <= r by NORMSP_0:def 4;
        hence thesis by B2,XXREAL_0:2;
      end; then
B5:   ||.seq.|| is bounded_above;
      for n be Nat holds -1 < ||.seq.||.n
      proof
        let n be Nat;
        ||.seq.n.|| = ||.seq.||.n by NORMSP_0:def 4;
        hence thesis;
      end; then
      ||.seq.|| is bounded_below; then
      consider seq1 be sequence of X such that
P1:     seq1 is subsequence of seq
      & seq1 is weakly-convergent by AS,Th813,B5;
      w-lim seq1 in X;
      hence thesis by P1;
    end;
    hence thesis;
end;
