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 Th30:
  for A,D being set for p being FinSequence of A for f being
  Function of A,D holds f*p is FinSequence of D
proof
  let A,D be set;
  let p be FinSequence of A;
  let f be Function of A,D;
  per cases;
  suppose D = {};
    then f*p = <*>D;
    hence thesis;
  end;
  suppose
A1: D <> {};
A2: rng p c= A by RELAT_1:def 19;
A3: rng(f*p) c= D by RELAT_1:def 19;
    dom f = A by A1,FUNCT_2:def 1;
    then f*p is FinSequence by A2,FINSEQ_1:16;
    hence thesis by A3,FINSEQ_1:def 4;
  end;
end;
