reserve
  X for non empty set,
  FX for Filter of X,
  SFX for Subset-Family of X;

theorem
  for X be non empty set,I be non empty set,
  M be (Filt X)-valued ManySortedSet of I holds
  for i be Element of I,F be Filter of X st F=M.i holds
  Filter_Intersection M is_filter-coarser_than F
   proof
     let X be non empty set,I be non empty set,
     M be (Filt X)-valued ManySortedSet of I;
     let i be Element of I,F be Filter of X;
     set FIM=Filter_Intersection M;
     assume
A1:  F=M.i;
     now
       let a be object;
       assume
A2:    a in Filter_Intersection M;
       i in I;
       then i in dom M by PARTFUN1:def 2;
       then M.i in rng M by FUNCT_1:def 3;
       hence a in F by A1,A2,SETFAM_1:def 1;
     end;
     then FIM c= F;
     hence FIM is_filter-coarser_than F;
 end;
