reserve x,y for set,
  n,m for Nat,
  r,s for Real;

theorem Th4:
  for f,g be FinSequence holds f,g are_fiberwise_equipotent iff
  ex P be Permutation of dom g st f = g*P
proof
  let f,g be FinSequence;
  thus f,g are_fiberwise_equipotent implies ex P be Permutation of dom g st f
  = g*P
  proof
    assume
A1: f,g are_fiberwise_equipotent;
    then dom f = dom g by Th3;
    hence thesis by A1,CLASSES1:80;
  end;
  given P be Permutation of dom g such that
A2: f = g*P;
  dom g = {} implies dom g = {};
  then rng P c= dom g & dom P = dom g by FUNCT_2:def 1,RELAT_1:def 19;
  then dom f = dom g by A2,RELAT_1:27;
  hence thesis by A2,CLASSES1:80;
end;
