reserve k,n,i,j for Nat;

theorem Th33:
  for n being Nat, K being Ring, p being Element of
  Permutations(n), f being FinSequence of K st n>=1 & len f=n holds f*p is
  FinSequence of K
proof
  let n be Nat, K be Ring, p be Element of Permutations(n), f be
  FinSequence of K;
  assume that
A1: n>=1 and
A2: len f=n;
  reconsider q=p as Function of Seg n,Seg n by MATRIX_1:def 12;
  p is bijective Function of Seg n,Seg n by MATRIX_1:def 12;
  then rng q = Seg n by FUNCT_2:def 3;
  then rng p c= dom f by A2,FINSEQ_1:def 3;
  then dom (f*p)=dom q by RELAT_1:27
    .=Seg n by A1,FUNCT_2:def 1;
  then reconsider h=f*p as FinSequence by FINSEQ_1:def 2;
  rng h c= rng f & rng f c= the carrier of K by FINSEQ_1:def 4,RELAT_1:26;
  then rng h c= the carrier of K;
  hence thesis by FINSEQ_1:def 4;
end;
