theorem Th5:
  N is triangle symmetric & S2 is convergent implies S2 is Cauchy
proof
  assume that
A1: N is triangle and
A2: N is symmetric;
   reconsider N as symmetric non empty MetrStruct by A2;
  assume
A3:  S2 is convergent;
  reconsider S2 as sequence of N;
  consider g being Element of N such that
A4: for r st 0<r ex n st for m st n<=m holds dist(S2.m,g)<r by A3;
  let r;
  assume 0<r;
  then consider n such that
A5: for m st n<=m holds dist(S2.m,g)<r/2 by A4,XREAL_1:215;
  take n;
  let m,m9 be Nat;
  assume that
A6: m>=n and
A7: m9>=n;
A8: dist(S2.m9,g)<r/2 by A5,A7;
  dist(S2.m,g)<r/2 by A5,A6;
  then
A9: dist(S2.m,g)+dist(g,S2.m9)<r/2+r/2 by A8,XREAL_1:8;
  dist(S2.m,S2.m9)<=dist(S2.m,g)+dist(g,S2.m9) by A1,METRIC_1:4;
  hence thesis by A9,XXREAL_0:2;
end;
