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;

theorem Th36:
  for f being Function of Seg i,Seg j st (j = 0 implies i = 0) & j <= len p
  holds p*f is FinSequence
proof
A0: i is Nat by TARSKI:1;
  let f be Function of Seg i,Seg j;
  assume j = 0 implies i = 0;
  then Seg j = {} implies Seg i = {};
  then
A1: dom f = Seg i by FUNCT_2:def 1;
  assume j <= len p;
  then rng f c= Seg j & Seg j c= Seg len p by FINSEQ_1:5,RELAT_1:def 19;
  then rng f c= Seg len p;
  then rng f c= dom p by FINSEQ_1:def 3;
  then dom(p*f) = dom f by RELAT_1:27;
  hence thesis by A0,A1,FINSEQ_1:def 2;
end;
