
theorem REFF1:
  for X be RealNormSpace holds X is Reflexive
 iff
   for f be Point of DualSp DualSp X
    ex x be Point of X st
      for g be Point of DualSp X holds f.g = g.x
proof
   let X be RealNormSpace;
   hereby assume X is Reflexive; then
A1: BidualFunc X is onto;
    thus for f be Point of DualSp DualSp X
      ex x be Point of X st
        for g be Point of DualSp X holds f.g = g.x
    proof
     let f be Point of DualSp DualSp X;
     consider x be object such that
A2:  x in dom(BidualFunc X) & f = (BidualFunc X).x by A1,FUNCT_1:def 3;
     reconsider x as Point of X by A2;
     take x;
     f = BiDual x by A2,Def2;
     hence thesis by Def1;
    end;
   end;
   assume B1: for f be Point of DualSp DualSp X
                ex x be Point of X st
                  for g be Point of DualSp X holds f.g = g.x;
   for v being object st v in the carrier of DualSp DualSp X
     ex s being object st s in the carrier of X
      & v = (BidualFunc X).s
   proof
    let v be object;
    assume v in the carrier of DualSp DualSp X; then
    reconsider f = v as Point of DualSp DualSp X;
    consider s be Point of X such that
B2: for g be Point of DualSp X holds f.g = g.s by B1;
    take s;
    thus s in the carrier of X;
    f = BiDual s by B2,Def1;
    hence v = (BidualFunc X).s by Def2;
   end; then
   BidualFunc X is onto by FUNCT_2:10;
   hence thesis;
end;
