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

theorem Th27:
  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 phi.(Y) c= Y
  holds
  (for u being Element of Q st u in H holds (curry (the multF of Q)).u in Y)
  &
  (for u being Element of Q st u in H holds (curry' (the multF of Q)).u in Y)
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 phi.(Y) c= Y;
  then A2: MltClos1(H,Y) c= Y by A1;
  thus for u being Element of Q st u in H holds
    (curry (the multF of Q)).u in Y
  proof
    let u be Element of Q;
    assume u in H;
    then (curry (the multF of Q)).u in MltClos1(H,Y) by Def37;
    hence thesis by A2;
  end;
    let u be Element of Q;
    assume u in H;
    then (curry' (the multF of Q)).u in MltClos1(H,Y) by Def37;
    hence thesis by A2;
end;
