reserve a,b,c,d,m,x,n,j,k,l for Nat,
  t,u,v,z for Integer,
  f,F for FinSequence of NAT;
reserve p,q,r,s for real number;
reserve a,b,c,d,m,x,n,k,l for Nat,
  t,z for Integer,
  f,F,G for FinSequence of REAL;
reserve q,r,s for real number;
reserve D for set;

theorem Th3: :: AFINSEQ_1:Lm1
  n+1 <= len f iff n+1 in dom f
  proof
    thus n+1 <= len f implies n+1 in dom f
    proof assume n+1 <= len f; then
      n < len f by NAT_1:13;
      then n+1 in Seg len f by Th2;
      hence n+1 in dom f by FINSEQ_1:def 3;
    end;
    assume n+1 in dom f;
    then n+1 in Seg len f by FINSEQ_1:def 3; then
    n < len f by Th2;
    hence n+1 <= len f by NAT_1:13;
  end;
