reserve k,n,i,j for Nat;

theorem Th31:
  for n being Nat, K being commutative Ring, p being Element of
  Permutations(n), f,g being FinSequence of K st len f=n & g=f*p holds f,g
  are_fiberwise_equipotent
proof
  let n be Nat, K be commutative Ring,
  p be Element of Permutations(n), f,g be
  FinSequence of K;
  assume that
A1: len f=n and
A2: g=f*p;
  reconsider fp=p as Function of Seg n,Seg n by MATRIX_1:def 12;
A3: p is Permutation of Seg n by MATRIX_1:def 12;
  then
A4: rng p=Seg n by FUNCT_2:def 3;
  rng fp = Seg n by A3,FUNCT_2:def 3;
  then rng p c= dom f by A1,FINSEQ_1:def 3;
  then
A5: dom (f*p)=dom fp by RELAT_1:27;
  dom f=Seg n by A1,FINSEQ_1:def 3;
  hence thesis by A2,A5,A4,CLASSES1:77;
end;
