reserve Q,Q1,Q2 for multLoop;
reserve x,y,z,w,u,v for Element of Q;

theorem Th29:
  for H being Subset of Q holds
  for phi being Function of bool Funcs(Q,Q),bool Funcs(Q,Q)
  st
  for X being Subset of Funcs(Q,Q) holds phi.X = MltClos1(H,X)
  holds
  for Y being Subset of Funcs(Q,Q)
  st
  Y is_a_fixpoint_of phi
  & for S be Subset of Funcs(Q,Q) st phi.S c= S holds Y c= S
  holds
  Y is composition-closed
  &
  Y is inverse-closed
proof
  let H be Subset of Q;
  let phi be Function of bool Funcs(Q,Q),bool Funcs(Q,Q);
  assume A1: for X being Subset of Funcs(Q,Q) holds phi.X = MltClos1(H,X);
  let Y be Subset of Funcs(Q,Q);
  assume Y is_a_fixpoint_of phi;
  then A2: Y in dom phi & Y = phi.(Y) & phi.(Y) = MltClos1(H,Y)
    by ABIAN:def 3,A1;
  assume A3: for S be Subset of Funcs(Q,Q) st phi.S c= S holds Y c= S;
  thus Y is composition-closed
  proof
    let f,g be Element of Y;
    assume A4: f in Y & g in Y;
    then f is Permutation of Q & g is Permutation of Q by Th28,A1,A3;
    hence f * g in Y by A2,Def37,A4;
  end;
    let f be Element of Y;
    assume A5: f in Y;
    then f is Permutation of Q by Th28,A1,A3;
    hence f" in Y by A2,Def37,A5;
end;
