reserve UA for Universal_Algebra,
  f, g for Function of UA, UA;
reserve I for set,
  A, B, C for ManySortedSet of I;

theorem
  for A, B be non-empty ManySortedSet of I for F be ManySortedFunction
  of A, B st F is "1-1" "onto" holds (F"")"" = F
proof
  let A, B be non-empty ManySortedSet of I;
  let F be ManySortedFunction of A, B;
  assume
A1: F is "1-1" "onto";
  now
    let i be object;
    assume
A2: i in I;
    then reconsider f = F.i as Function of A.i, B.i by PBOOLE:def 15;
    reconsider f9 = (F"").i as Function of B.i, A.i by A2,PBOOLE:def 15;
    f is one-to-one by A1,A2,MSUALG_3:1;
    then
A3: (f")" = f by FUNCT_1:43;
    F"" is "1-1" "onto" by A1,Th12;
    then (F"")"".i = f9" by A2,MSUALG_3:def 4;
    hence (F"")"".i = F.i by A1,A2,A3,MSUALG_3:def 4;
  end;
  hence thesis by PBOOLE:3;
end;
