
theorem CLOSE1:
  for X be RealNormSpace,
      Y be Subset of X,
      v be VECTOR of X
  st Y is closed
   & for e be Real st 0 < e
     holds ex w be VECTOR of X st w in Y & ||.v-w.|| <= e
  holds v in Y
  proof
    let X be RealNormSpace,
        Y be Subset of X,
        v be VECTOR of X;
    assume that
    A1: Y is closed and
    A2: for e be Real st 0 < e
      holds ex w be VECTOR of X st w in Y & ||.v-w.|| <= e;
    assume
    A3: not v in Y;
    reconsider Z = Y` as Subset of TopSpaceNorm X;
    A4: Z is open by A1,NORMSP_2:16;
    v in (the carrier of X) \ Y by A3,XBOOLE_0:def 5; then
    v in Z by SUBSET_1:def 4; then
    consider e be Real such that
    A5: e > 0 & {y where y is Point of X: ||.v-y.|| < e} c= Z by A4,NORMSP_2:7;
    consider w be VECTOR of X such that
    A6: w in Y & ||.v-w.|| <= e/2 by A2,A5;
    e/2 < e by A5,XREAL_1:216; then
    ||.v-w.|| < e by A6,XXREAL_0:2; then
    w in {y where y is Point of X: ||.v-y.|| < e}; then
    w in Y` by A5; then
    w in (the carrier of X) \ Y by SUBSET_1:def 4;
    hence contradiction by A6,XBOOLE_0:def 5;
  end;
