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
  for x being set, f being Function st x in dom f holds f*<*x*> = <*f.x*>
proof
  let x be set, f be Function;
  assume A1: x in dom f;
  then reconsider D = dom f, E = rng f as non empty set by FUNCT_1:3;
  rng <*x*> = {x} by FINSEQ_1:38;
  then rng <*x*> c= D by A1,ZFMISC_1:31;
  then reconsider p = <*x*> as FinSequence of D by FINSEQ_1:def 4;
  reconsider f as Function of D, E by FUNCT_2:def 1,RELSET_1:4;
  reconsider q = f*p as FinSequence of E by Th30;
A2: p.1 = x;
A3: len q = len p by Th31 .= 1 by FINSEQ_1:39;
  then 1 in Seg len q;
  then 1 in dom q by FINSEQ_1:def 3;
  then q.1 = f.x by A2,FUNCT_1:12;
  hence thesis by A3,FINSEQ_1:40;
end;
