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;
