
theorem Th27:
  for N be Subset of REAL, M be Subset of R^1 st N = M holds
    (for F be Subset-Family of REAL
       st F is Cover of N
        & (for P be Subset of REAL st P in F holds P is open)
     holds
       ex G be Subset-Family of REAL
         st G c= F & G is Cover of N & G is finite)
  iff
    (for F1 be Subset-Family of R^1 st F1 is Cover of M & F1 is open
     holds
       ex G1 be Subset-Family of R^1
         st G1 c= F1 & G1 is Cover of M & G1 is finite)
  proof
    let N be Subset of REAL, M be Subset of R^1;
    assume
    A1: N = M;
    hereby
      assume
      A2: for F be Subset-Family of REAL
            st F is Cover of N
             & (for P be Subset of REAL st P in F holds P is open)
          holds
            ex G be Subset-Family of REAL
              st G c= F & G is Cover of N & G is finite;
      thus for F1 be Subset-Family of R^1
        st F1 is Cover of M & F1 is open holds
        ex G1 be Subset-Family of R^1
          st G1 c= F1 & G1 is Cover of M & G1 is finite
      proof
        let F1 be Subset-Family of R^1;
        assume
        A3: F1 is Cover of M & F1 is open;
        reconsider F = F1 as Subset-Family of REAL;
        for P be Subset of REAL st P in F holds P is open
        proof
          let P be Subset of REAL;
          assume
          A4: P in F;
          reconsider P1 = P as Subset of R^1;
          P1 is open by A3,A4;
          hence P is open by BORSUK_5:39;
        end; then
        consider G be Subset-Family of REAL such that
        A5: G c= F & G is Cover of N & G is finite by A1,A2,A3;
        reconsider G1 = G as Subset-Family of R^1;
        take G1;
        thus thesis by A1,A5;
      end;
    end;
    assume
    A6: for F1 be Subset-Family of R^1
          st F1 is Cover of M & F1 is open holds
          ex G1 be Subset-Family of R^1
            st G1 c= F1 & G1 is Cover of M & G1 is finite;
      let F be Subset-Family of REAL;
      assume that
      A7: F is Cover of N and
      A8: for P be Subset of REAL st P in F holds P is open;
      reconsider F1 = F as Subset-Family of R^1;
      for P1 be Subset of R^1 st P1 in F1 holds P1 is open
      proof
        let P1 be Subset of R^1;
        assume
        A9: P1 in F1;
        reconsider P = P1 as Subset of REAL;
        P is open by A8,A9;
        hence P1 is open by BORSUK_5:39;
      end; then
      F1 is open; then
      consider G1 be Subset-Family of R^1 such that
      A10: G1 c= F1 & G1 is Cover of M & G1 is finite by A1,A6,A7;
      reconsider G = G1 as Subset-Family of REAL;
      take G;
      thus thesis by A1,A10;
  end;
