
theorem Th7:
for I be closed_interval Subset of REAL, E be Subset of RNS_Real
 st I = E holds E is compact
proof
    let I be closed_interval Subset of REAL, E be Subset of RNS_Real;
    assume
A1:  I = E;

    for s1 being sequence of RNS_Real st rng s1 c= E holds
     ex s2 being sequence of RNS_Real st
      s2 is subsequence of s1 & s2 is convergent & lim s2 in E
    proof
     let s1 be sequence of RNS_Real;
     assume
A2:   rng s1 c= E;

     reconsider t1=s1 as Real_Sequence;

     consider t2 being Real_Sequence such that
A3:   t2 is subsequence of t1 & t2 is convergent
    & lim t2 in I by A1,A2,RCOMP_1:def 3;
     consider N1 being increasing sequence of NAT such that
A4:   t2 = t1*N1 by A3,VALUED_0:def 17;

     reconsider s2=s1*N1 as sequence of RNS_Real;

     take s2;
     thus s2 is subsequence of s1;
     reconsider s0=lim t2 as Point of RNS_Real by XREAL_0:def 1;

A5:  now let r be Real;
      assume 0 < r; then
      consider n be Nat such that
A6:    for m be Nat st n<=m holds |.t2.m-lim t2.|< r by A3,SEQ_2:def 7;

      take n;
      let m be Nat;
      assume n<=m; then
A7:   |.t2.m-lim t2.|< r by A6;
      t2.m-lim t2 =s2.m -s0 by A4,DUALSP03:4;
      hence ||.s2.m-s0.||< r by A7,EUCLID:def 2;
     end;
     hence s2 is convergent;
     hence lim s2 in E by A1,A3,A5,NORMSP_1:def 7;
    end;
    hence E is compact;
end;
