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 Th39:
  for f being Function of Seg i,Seg i st rng f = Seg i & i <= len
  p & q = p*f holds len q = i
proof
  let f be Function of Seg i,Seg i;
  assume rng f = Seg i & i <= len p;
  then rng f c= Seg len p by FINSEQ_1:5;
  then rng f c= dom p by FINSEQ_1:def 3;
  then
A1: dom(p*f) = dom f by RELAT_1:27;
  i is Element of NAT & dom f = Seg i by FUNCT_2:52,ORDINAL1:def 12;
  hence thesis by A1,FINSEQ_1:def 3;
end;
