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 F being Function of [:D,D9:],E for p being FinSequence of D9 st p
  = <*d19,d29*> holds F[;](d,p) = <*F.(d,d19),F.(d,d29)*>
proof
  let F be Function of [:D,D9:],E;
  let p be FinSequence of D9 such that
A1: p = <*d19,d29*>;
A2: p.2 = d29 by A1;
  reconsider r = F[;](d,p) as FinSequence of E by Th75;
  len p = 2 by A1,FINSEQ_1:44;
  then
A3: len r = 2 by Th76;
  then 2 in Seg len r;
  then 2 in dom r by FINSEQ_1:def 3;
  then
A4: r.2 = F.(d,d29) by A2,FUNCOP_1:32;
  1 in Seg len r by A3;
  then
A5: 1 in dom r by FINSEQ_1:def 3;
  p.1 = d19 by A1;
  then r.1 = F.(d,d19) by A5,FUNCOP_1:32;
  hence thesis by A3,A4,FINSEQ_1:44;
end;
