reserve i,j,n,k,m for Nat,
     a,b,x,y,z for object,
     F,G for FinSequence-yielding FinSequence,
     f,g,p,q for FinSequence,
     X,Y for set,
     D for non empty set;

theorem Th44:
  for f be finite Function, p be Permutation of dom f holds
    f is with_evenly_repeated_values iff f*p is with_evenly_repeated_values
proof
  let f be finite Function, p be Permutation of dom f;
  set fp=f*p;
A1: dom p = dom f = rng p by FUNCT_2:52,def 3;
  then
A2: dom fp = dom f by RELAT_1:27;
  thus f is with_evenly_repeated_values implies
     (f*p) is with_evenly_repeated_values
  proof
    assume
A3:   f is with_evenly_repeated_values;
    let y be object;
    card (fp"{y}) = card (f"{y}) by A2,A1,CLASSES1:77,78;
    hence thesis by A3;
  end;
  assume
A4: f*p is with_evenly_repeated_values;
  let y be object;
  card (fp"{y}) = card (f"{y}) by A2,A1,CLASSES1:77,78;
  hence thesis by A4;
end;
