reserve E, F, G,S,T,W,Y for RealNormSpace;
reserve f,f1,f2 for PartFunc of S,T;
reserve Z for Subset of S;
reserve i,n for Nat;

theorem Th64:
  for F be non trivial RealBanachSpace
  holds R_NormSpace_of_BoundedLinearOperators(F,F)
      is non trivial RealBanachSpace
proof
  let F be non trivial RealBanachSpace;
  set F1 = R_NormSpace_of_BoundedLinearOperators (F,F);

  not for x be object st x in [#]F holds x = 0.F
  proof
    assume A1: for x be object st x in [#]F holds x = 0.F;

    for x, y be object st x in [#]F & y in [#]F holds x = y
    proof
      let x, y be object;
      assume x in [#]F & y in [#]F;
      then x = 0.F & y = 0.F by A1;
      hence thesis;
    end;
    hence contradiction by ZFMISC_1:def 10;
  end;
  then consider y be object such that
  A2: y in [#]F & y <> 0.F;

  reconsider L1 = id [#]F as Lipschitzian LinearOperator of F,F by LOPBAN_2:3;
  reconsider L2 = L1 as Point of F1 by LOPBAN_1:def 9;
  reconsider Z0 = 0.F1 as Lipschitzian LinearOperator of F,F by LOPBAN_1:def 9;

  0.F1 = [#]F --> 0.F by LOPBAN_1:31; then
  (0.F1).y = 0.F by A2,FUNCOP_1:7; then
  L2 <> 0.F1 by A2,FUNCT_1:17;
  hence thesis by ZFMISC_1:def 10;
end;
