reserve i,j,k,l for natural Number;
reserve A for set, a,b,x,x1,x2,x3 for object;
reserve D,D9,E for non empty set;
reserve d,d1,d2,d3 for Element of D;
reserve d9,d19,d29,d39 for Element of D9;
reserve p,q,r for FinSequence;
reserve s for Element of D*;
reserve m,n for Nat,
  s,w for FinSequence of NAT;

theorem Th130:
  for A being set, i being Nat, p being FinSequence
  holds p in i-tuples_on A iff len p = i & rng p c= A
proof
  let A be set, i be Nat, p be FinSequence;
  hereby
    assume p in i-tuples_on A;
    then ex q being Element of A* st p = q & len q = i;
    hence len p = i & rng p c= A by FINSEQ_1:def 4;
  end;
  assume
A1: len p = i;
  assume rng p c= A;
  then p is FinSequence of A by FINSEQ_1:def 4;
  then p in A* by FINSEQ_1:def 11;
  hence thesis by A1;
end;
