
theorem RNS8:
  for seq be Real_Sequence, seq1 be sequence of RNS_Real
    st seq = seq1 holds seq is convergent iff seq1 is convergent
proof
  let seq be Real_Sequence, seq1 be sequence of RNS_Real;
  assume AS: seq = seq1;
  hereby assume P1: seq is convergent;
  reconsider s1=lim seq as Point of RNS_Real by XREAL_0:def 1;
  now let p be Real;
    assume 0 < p; then
    consider n be Nat such that
P2:   for m be Nat st n <= m holds |.seq.m - lim seq .| < p
        by P1,SEQ_2:def 7;
    take n;
    hereby let m be Nat;
      assume n <= m; then
P3:   |.seq.m - lim seq .| < p by P2;
      seq.m - lim seq = seq1.m - s1 by AS,RNS4;
      hence ||.seq1.m - s1.|| < p by P3,EUCLID:def 2;
    end;
  end;
  hence seq1 is convergent;
end;
  assume P4: seq1 is convergent;
  reconsider s1=lim seq1 as Real;
  now let p be Real;
    assume 0 < p; then
    consider n be Nat such that
P2:   for m be Nat st n <= m holds ||.seq1.m - lim seq1 .|| < p
        by P4,NORMSP_1:def 7;
    take n;
    hereby let m be Nat;
      assume n <= m; then
P3:   ||.seq1.m - lim seq1 .|| < p by P2;
      seq1.m - lim seq1 = seq.m - s1 by AS,RNS4;
      hence |.seq.m - s1.| < p by P3,EUCLID:def 2;
    end;
  end;
  hence seq is convergent;
end;
