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 D for q
  being FinSequence of D9 st p = <*d1*> & q = <*d19*> holds F.:(p,q) = <*F.(d1,
  d19)*>
proof
  let F be Function of [:D,D9:],E;
  let p be FinSequence of D;
  let q be FinSequence of D9 such that
A1: p = <*d1*> & q = <*d19*>;
A2: p.1 = d1 & q.1 = d19 by A1;
  reconsider r = F.:(p,q) as FinSequence of E by Th68;
  len p = 1 & len q = 1 by A1,FINSEQ_1:39;
  then
A3: len r = 1 by Th70;
  then 1 in Seg len r;
  then 1 in dom r by FINSEQ_1:def 3;
  then r.1 = F.(d1,d19) by A2,FUNCOP_1:22;
  hence thesis by A3,FINSEQ_1:40;
end;
