
:: Baire category theorem - Banach space version
  for X be RealBanachSpace, Y be SetSequence of X st union rng Y =
the carrier of X & (for n be Nat holds Y.n is closed)
 ex n0 be Nat, r be Real, x0 be Point of X
    st 0 < r & Ball (x0,r) c= Y.n0
proof
  let X be RealBanachSpace, Y be SetSequence of X;
  assume that
A1: union rng Y = the carrier of X and
A2: for n be Nat holds Y.n is closed;
  now
    let n be Nat;
    reconsider Yn = Y.n as Subset of TopSpaceNorm X;
    Y.n is closed by A2;
    then Yn is closed by NORMSP_2:15;
    then Yn` is open by TOPS_1:3;
    hence (Y.n)` in Family_open_set MetricSpaceNorm X by PRE_TOPC:def 2;
  end;
  then consider
  n0 be Nat, r be Real, xx0 be Point of MetricSpaceNorm X
  such that
A3: 0 < r & Ball(xx0,r) c= Y.n0 by A1,NORMSP_2:1;
  consider x0 be Point of X such that
  x0 = xx0 and
A4: Ball(xx0,r) = {x where x is Point of X:||.x0-x.|| < r} by NORMSP_2:2;
  Ball (x0,r) = {x where x is Point of X : ||.x0-x.|| < r };
  hence thesis by A3,A4;
end;
