
theorem
  for S1 be sequence of RealSpace, seq be Real_Sequence,
      g be Real, g1 be Element of RealSpace st S1 = seq & g = g1
  holds
      (seq is convergent & lim seq = g)
    iff
      (S1 is convergent & lim S1 = g1)
  proof
    let S1 be sequence of RealSpace, seq be Real_Sequence,
        g be Real, g1 be Element of RealSpace;
    assume AS: S1=seq & g=g1;
    hereby
      assume seq is convergent & lim seq = g; then
      for p be Real st 0 < p
      ex n be Nat st for m be Nat st n <= m holds |.seq.m - g.| < p
        by SEQ_2:def 7; then
      A1: for p be Real st 0 < p
          ex n be Nat st for m be Nat st n <= m holds dist(S1.m,g1) < p
            by AS,Th23;
      hence S1 is convergent;
      hence lim S1 = g1 by A1,TBSP_1:def 3;
    end;
    assume S1 is convergent & lim S1 = g1; then
    for p be Real st 0 < p
    ex n be Nat st for m be Nat st n <= m holds dist(S1.m,g1) < p
      by TBSP_1:def 3; then
A2: for p be Real st 0 < p
      ex n be Nat st for m be Nat st n <= m holds |.seq.m - g.| < p
        by AS,Th23;
    hence seq is convergent by SEQ_2:def 6;
    hence lim seq = g by A2,SEQ_2:def 7;
  end;
