
theorem Th7:
  for a,b being Real, S1 being sequence of
Closed-Interval-MSpace(a,b), S being sequence of RealSpace st S=S1 & a<=b holds
(S is convergent iff S1 is convergent)& (S is convergent implies lim S=lim S1)
proof
  let a,b be Real, S1 be sequence of Closed-Interval-MSpace(a,b), S be
  sequence of RealSpace;
  assume that
A1: S=S1 and
A2: a<=b;
  reconsider P=[.a,b.] as Subset of R^1 by TOPMETR:17;
A3: the carrier of Closed-Interval-MSpace(a,b)=[.a,b.] by A2,TOPMETR:10;
A4: S is convergent implies S1 is convergent
  proof
A5: for m being Nat holds S.m in [.a,b.]
    proof
      let m be Nat;
A6:   m in NAT by ORDINAL1:def 12;
      dom S1=NAT by FUNCT_2:def 1;
      then S1.m in rng S1 by FUNCT_1:def 3,A6;
      hence thesis by A1,A3;
    end;
A7: P is closed by TREAL_1:1;
    assume
A8: S is convergent;
    then consider x0 being Element of RealSpace such that
A9: for r being Real st r>0 ex n being Nat st for m being
    Nat st n<=m holds dist(S.m,x0)<r by TBSP_1:def 2;
    x0=lim S by A8,A9,TBSP_1:def 3;
    then reconsider x1=x0 as Element of Closed-Interval-MSpace(a,b) by A3,A8,A5
,A7,Th1,TOPMETR:def 6;
    for r being Real st r>0 ex n being Nat st for m being
    Nat st n<=m holds dist(S1.m,x1)<r
    proof
      let r be Real;
      assume r>0;
      then consider n0 being Nat such that
A10:   for m being Nat st n0<=m holds dist(S.m,x0)<r by A9;
      for m being Nat st n0<=m holds dist(S1.m,x1)<r
      proof
        let m be Nat;
        assume
A11:    n0<=m;
        dist(S1.m,x1) =(the distance of Closed-Interval-MSpace(a,b)).(S1.
        m,x1) by METRIC_1:def 1
          .=(the distance of RealSpace).(S1.m,x1) by TOPMETR:def 1
          .=dist(S.m,x0) by A1,METRIC_1:def 1;
        hence thesis by A10,A11;
      end;
      hence thesis;
    end;
    hence thesis by TBSP_1:def 2;
  end;
  S1 is convergent implies S is convergent
  proof
    assume S1 is convergent;
    then consider x0 being Element of Closed-Interval-MSpace(a,b) such that
A12: for r being Real st r>0 ex n being Nat st for m being
    Nat st n<=m holds dist(S1.m,x0)<r by TBSP_1:def 2;
    x0 in [.a,b.] by A3;
    then reconsider x1=x0 as Element of RealSpace by METRIC_1:def 13;
    for r being Real st r>0 ex n being Nat st for m being
    Nat st n<=m holds dist(S.m,x1)<r
    proof
      let r be Real;
      assume r>0;
      then consider n0 being Nat such that
A13:  for m being Nat st n0<=m holds dist(S1.m,x0)<r by A12;
      for m being Nat st n0<=m holds dist(S.m,x1)<r
      proof
        let m be Nat;
        assume
A14:    n0<=m;
        dist(S1.m,x0) =(the distance of Closed-Interval-MSpace(a,b)).(S1.
        m,x0) by METRIC_1:def 1
          .=(the distance of RealSpace).(S1.m,x0) by TOPMETR:def 1
          .=dist(S.m,x1) by A1,METRIC_1:def 1;
        hence thesis by A13,A14;
      end;
      hence thesis;
    end;
    hence thesis by TBSP_1:def 2;
  end;
  hence S is convergent iff S1 is convergent by A4;
  thus S is convergent implies lim S=lim S1
  proof
    lim S1 in the carrier of Closed-Interval-MSpace(a,b);
    then reconsider t0=lim S1 as Point of RealSpace by A3,METRIC_1:def 13;
    assume
A15: S is convergent;
    for r1 being Real st 0 < r1 ex m1 being Nat st for n1
    being Nat st m1 <= n1 holds dist(S.n1,t0) < r1
    proof
      let r1 being Real;
      assume 0<r1;
      then consider m1 being Nat such that
A16:  for n1 being Nat st m1 <= n1 holds dist(S1.n1,lim S1
      ) < r1 by A4,A15,TBSP_1:def 3;
      for n1 being Nat st m1 <= n1 holds dist(S.n1,t0) < r1
      proof
        let n1 be Nat;
        assume
A17:    m1 <= n1;
        dist(S1.n1,lim S1) =(the distance of Closed-Interval-MSpace(a,b))
        .(S1.n1,lim S1) by METRIC_1:def 1
          .=(the distance of RealSpace).(S1.n1,lim S1) by TOPMETR:def 1
          .=dist(S.n1,t0) by A1,METRIC_1:def 1;
        hence thesis by A16,A17;
      end;
      hence thesis;
    end;
    hence thesis by A15,TBSP_1:def 3;
  end;
end;
