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 Th34:
  for p being FinSequence of D for f being Function of D,D9 st p =
  <*x1,x2*> holds f*p = <*f.x1,f.x2*>
proof
  let p be FinSequence of D;
  let f be Function of D,D9 such that
A1: p = <*x1,x2*>;
A2: p.2 = x2 by A1;
  reconsider q = f*p as FinSequence of D9 by Th30;
  len p = 2 by A1,FINSEQ_1:44;
  then
A3: len q = 2 by Th31;
  then 2 in Seg len q;
  then 2 in dom q by FINSEQ_1:def 3;
  then
A4: q.2 = f.x2 by A2,FUNCT_1:12;
  1 in Seg len q by A3;
  then
A5: 1 in dom q by FINSEQ_1:def 3;
  p.1 = x1 by A1;
  then q.1 = f.x1 by A5,FUNCT_1:12;
  hence thesis by A3,A4,FINSEQ_1:44;
end;
