reserve X for RealNormSpace;

theorem Th26:
  for X be RealNormSpace, S be sequence of TopSpaceNorm X, St be
sequence of LinearTopSpaceNorm X, x be Point of TopSpaceNorm X, xt be Point of
  LinearTopSpaceNorm X st S=St & x=xt holds St is_convergent_to xt iff S
  is_convergent_to x
proof
  let X be RealNormSpace, S be sequence of TopSpaceNorm X, St be sequence of
  LinearTopSpaceNorm X, x be Point of TopSpaceNorm X, xt be Point of
  LinearTopSpaceNorm X;
  assume that
A1: S=St and
A2: x=xt;
A3: now
    assume
A4: S is_convergent_to x;
    now
      let U1t be Subset of LinearTopSpaceNorm X such that
A5:   U1t is open and
A6:   xt in U1t;
      reconsider U1=U1t as open Subset of TopSpaceNorm X by A5,Def4,Th20;
      consider n being Nat such that
A7:   for m being Nat st n <= m holds S.m in U1 by A2,A4,A6,
FRECHET:def 3;
      take n;
      let m be Nat;
      assume n <= m;
      hence St.m in U1t by A1,A7;
    end;
    hence St is_convergent_to xt by FRECHET:def 3;
  end;
  now
    assume
A8: St is_convergent_to xt;
    now
      let U1 be Subset of TopSpaceNorm X such that
A9:   U1 is open and
A10:  x in U1;
      reconsider U1t=U1 as open Subset of LinearTopSpaceNorm X by A9,Def4,Th20;
      consider n being Nat such that
A11:  for m being Nat st n <= m holds St.m in U1t by A2,A8,A10,
FRECHET:def 3;
      take n;
      let m be Nat;
      assume n <= m;
      hence S.m in U1 by A1,A11;
    end;
    hence S is_convergent_to x by FRECHET:def 3;
  end;
  hence thesis by A3;
end;
