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

theorem Th34:
  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
  Mlt H is_a_fixpoint_of phi
  &
  for S be Subset of Funcs(Q,Q) st phi.S c= S holds Mlt H c= S
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);
  Mlt H in bool Funcs(Q,Q)& phi is quasi_total;
  then A2: Mlt H in dom phi by FUNCT_2:def 1;
  A3: phi.(Mlt H) c= Mlt H
  proof
    let f be object;
    assume f in phi.(Mlt H);
    then f in MltClos1(H,Mlt H) by A1;
    then PQ[Q,H,Mlt H,f] by Def37;
    hence thesis by Th33,Th32,Def34,Def35;
  end;
  A4: for S be Subset of Funcs(Q,Q) st phi.S c= S holds Mlt H c= S
  proof
    let S be Subset of Funcs(Q,Q);
    assume A5: phi.S c= S;
    set SP = {f where f is Permutation of Q : f in S};
    A6: SP c= S
    proof
      let g be object;
      assume g in SP;
      then ex f being Permutation of Q st g = f & f in S;
      hence thesis;
    end;
    S c= Funcs(the carrier of Q,the carrier of Q);
    then SP c= Funcs(the carrier of Q,the carrier of Q) by A6;
    then reconsider SP as Subset of Funcs(the carrier of Q,the carrier of Q);
    A7: for f being Element of SP st f in SP holds
      f is Permutation of the carrier of Q
    proof
      let f be Element of SP;
      assume f in SP;
      then ex g be Permutation of Q st f = g & g in S;
      hence thesis;
    end;
    for f,g being Element of SP st f in SP & g in SP holds f*g in SP
    proof
      let f,g be Element of SP;
      assume A8: f in SP & g in SP;
      reconsider f,g as Permutation of the carrier of Q by A7,A8;
      f*g in MltClos1(H,S) by Def37,A8,A6;
      then f*g in phi.S by A1;
      hence thesis by A5;
    end;
    then A9: SP is composition-closed ;
    for f being Element of SP st f in SP holds f" in SP
    proof
      let f be Element of SP;
      assume A10: f in SP;
      then f in S & f is Permutation of Q by A6,A7;
      then f" in MltClos1(H,S) by Def37;
      then A11: f" in phi.S by A1;
      reconsider f as Permutation of Q by A10,A7;
      f" is Permutation of Q;
      hence thesis by A11,A5;
    end;
    then SP is inverse-closed;
    then reconsider SP as composition-closed inverse-closed Subset of
     Funcs(Q,Q) by A9;
    for u being Element of Q st u in H holds
    (curry (the multF of Q)).u in SP
    &
    (curry' (the multF of Q)).u in SP
    proof
      let u be Element of Q;
      assume A12: u in H;
      then (curry (the multF of Q)).u in MltClos1(H,S) by Def37;
      then A13: (curry (the multF of Q)).u in phi.(S) by A1;
      (curry (the multF of Q)).u is Permutation of Q by Th30;
      hence (curry (the multF of Q)).u in SP by A13,A5;
      (curry' (the multF of Q)).u in MltClos1(H,S) by Def37,A12;
      then A14: (curry' (the multF of Q)).u in phi.(S) by A1;
      (curry' (the multF of Q)).u is Permutation of Q by Th31;
      hence (curry' (the multF of Q)).u in SP by A14,A5;
    end;
    then H left-right-mult-closed SP;
    then Mlt H c= SP by Def38;
    hence thesis by A6;
  end;
  Mlt H c= phi.(Mlt H)
  proof
    for f,g being Element of phi.(Mlt H) st f in phi.(Mlt H) & g in phi.(Mlt H)
      holds f*g in phi.(Mlt H)
    proof
      let f,g be Element of phi.(Mlt H);
      assume A15: f in phi.(Mlt H) & g in phi.(Mlt H);
      then f is Permutation of Q & g is Permutation of Q by Th28,A1,A4,A3;
      then f * g in MltClos1(H,Mlt H) by Def37,A15,A3;
      hence thesis by A1;
    end;
    then A16: phi.(Mlt H) is composition-closed;
    for f being Element of phi.(Mlt H) st
      f in phi.(Mlt H) holds f" in phi.(Mlt H)
    proof
      let f be Element of phi.(Mlt H);
      assume A17: f in phi.(Mlt H);
      then f is Permutation of Q by A3,Th28,A1,A4;
      then f" in MltClos1(H,Mlt H) by Def37,A17,A3;
      hence thesis by A1;
    end;
    then phi.(Mlt H) is inverse-closed;
    then reconsider S = phi.(Mlt H) as composition-closed inverse-closed
    Subset of Funcs(Q,Q) by A16;
    for u being Element of Q st u in H holds
    (curry (the multF of Q)).u in S &
    (curry' (the multF of Q)).u in S
    proof
      let u be Element of Q;
      assume u in H;
      then (curry (the multF of Q)).u in MltClos1(H,Mlt H)
      & (curry' (the multF of Q)).u in MltClos1(H,Mlt H)
      by Def37;
      hence thesis by A1;
    end;
    then H left-right-mult-closed S;
    hence thesis by Def38;
  end;
  then Mlt H = phi.(Mlt H) by A3;
  hence thesis by A4,ABIAN:def 3,A2;
end;
