reserve M for non empty MetrSpace,
  c,g1,g2 for Element of M;
reserve N for non empty MetrStruct,
  w for Element of N,
  G for Subset-Family of N,
  C for Subset of N;
reserve R for Reflexive non empty MetrStruct;
reserve T for Reflexive symmetric triangle non empty MetrStruct,
  t1 for Element of T,
  Y for Subset-Family of T,
  P for Subset of T;
reserve f for Function,
  n,m,p,n1,n2,k for Nat,
  r,s,L for Real,
  x,y for set;
reserve S1 for sequence of M,
  S2 for sequence of N;

theorem
  f is sequence of N iff dom f = NAT & for n holds f.n is Element of N
proof
  thus f is sequence of N implies
    dom f = NAT & for n holds f.n is Element of N
  by Lm1,ORDINAL1:def 12;
  assume that
A1: dom f = NAT and
A2: for n holds f.n is Element of N;
  for x holds x in NAT implies f.x is Element of N by A2;
  hence thesis by A1,Lm1;
end;
