reserve k,n,m,l,p for Nat;
reserve n0,m0 for non zero Nat;
reserve f for FinSequence;
reserve x,X,Y for set;

theorem Th9:
  for f1 being FinSequence of NAT, f2 being FinSequence of X st
  rng f1 c= dom f2 holds f2*f1 is FinSequence of X
proof
  let f1 be FinSequence of NAT;
  let f2 be FinSequence of X;
  consider n be Nat such that
A1: dom f1 = Seg n by FINSEQ_1:def 2;
  assume rng f1 c= dom f2;
  then dom(f2*f1) = Seg n by A1,RELAT_1:27;
  then
A2: f2*f1 is FinSequence by FINSEQ_1:def 2;
  rng(f2*f1) c= X;
  hence f2*f1 is FinSequence of X by A2,FINSEQ_1:def 4;
end;
