reserve p,q,r for FinSequence;
reserve u,v,x,y,y1,y2,z for object, A,D,X,Y for set;
reserve i,j,k,l,m,n for Nat;

theorem Th47:
  p | Seg n = p iff len p <= n
proof
  thus p | Seg n = p implies len p <= n by FINSEQ_1:86;
  assume len p <= n;
  then Seg len p c= Seg n by FINSEQ_1:5;
  then dom p c= Seg n by FINSEQ_1:def 3;
  then
A1: dom p = dom p /\ Seg n by XBOOLE_1:28;
  for x being object st x in dom p holds p.x = p.x;
  hence thesis by A1,FUNCT_1:46;
end;
