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 Th81:
  for F being Function of [:D,D9:],E for p being FinSequence of D
  holds F[:](p,d9) is FinSequence of E
proof
  let F be Function of [:D,D9:],E;
  let p be FinSequence of D;
A1: rng(F[:](p,d9)) c= rng F by RELAT_1:26;
  rng p c= D by FINSEQ_1:def 4;
  then [:rng p,{d9}:] c= [:D,D9:] by ZFMISC_1:96;
  then [:rng p,{d9}:] c= dom F by FUNCT_2:def 1;
  then
A2: F[:](p,d9) is FinSequence by Th66;
  rng F c= E by RELAT_1:def 19;
  then rng(F[:](p,d9)) c= E by A1;
  hence thesis by A2,FINSEQ_1:def 4;
end;
