reserve I,J for set,i,j,x for object,
  S for non empty ManySortedSign;

theorem
  for F be Function st dom F = {{}} holds Commute F = F
proof
  let F be Function;
  assume
A1: dom F = {{}};
A2: dom (Commute F) = {{}}
  proof
    thus dom (Commute F) c= {{}}
    proof
      let x be object;
      assume x in dom (Commute F);
      then ex f be Function st f in dom F & x = commute f by Def1;
      then x = {} by A1,FUNCT_6:58,TARSKI:def 1;
      hence thesis by TARSKI:def 1;
    end;
    let x be object;
    assume x in {{}};
    then
A3: x = {} by TARSKI:def 1;
    {} in dom F by A1,TARSKI:def 1;
    hence thesis by A3,Def1,FUNCT_6:58;
  end;
  for x being object st x in {{}} holds (Commute F).x = F.x
  proof
    let x be object;
    assume
A4: x in {{}};
    then x = {} by TARSKI:def 1;
    hence thesis by A2,A4,Def1,FUNCT_6:58;
  end;
  hence thesis by A1,A2,FUNCT_1:2;
end;
