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

theorem Th26:
  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
  phi is c=-monotone
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 a1, b1 be set such that
  A2: a1 in dom phi & b1 in dom phi & a1 c= b1;
  thus phi.a1 c= phi.b1
  proof
    reconsider a2 = a1, b2=b1 as Subset of Funcs(Q,Q)
        by A2,FUNCT_2:def 1;
    let f be object;
    assume f in phi.a1;
    then f in MltClos1(H,a2) by A1;
    then PQ[Q,H,a2,f]  by Def37;
    then f in MltClos1(H,b2) by A2,Def37;
    hence thesis by A1;
  end;
end;
