
theorem
  for L being Semilattice for X being upper non empty Subset of L holds
  X is Filter of L iff subrelstr X is meet-inheriting
proof
  let L be Semilattice, X be upper non empty Subset of L;
  set S = subrelstr X;
A1: the carrier of S = X by YELLOW_0:def 15;
  hereby
    assume
A2: X is Filter of L;
    thus S is meet-inheriting
    proof
      let x,y be Element of L;
      assume that
A3:   x in the carrier of S and
A4:   y in the carrier of S and
A5:   ex_inf_of {x,y},L;
      consider z being Element of L such that
A6:   z in X and
A7:   x >= z and
A8:   y >= z by A1,A2,A3,A4,Def2;
      z is_<=_than {x,y} by A7,A8,YELLOW_0:8;
      then z <= inf {x,y} by A5,YELLOW_0:def 10;
      hence thesis by A1,A6,Def20;
    end;
  end;
  assume
A9: for x,y being Element of L st
  x in the carrier of S & y in the carrier of S & ex_inf_of {x,y},L
  holds inf {x,y} in the carrier of S;
  X is filtered
  proof
    let x,y be Element of L;
    assume that
A10: x in X and
A11: y in X;
    take z = inf {x,y};
A12: z = x "/\" y by YELLOW_0:40;
    ex_inf_of {x,y},L by YELLOW_0:21;
    hence z in X & z <= x & z <= y by A1,A9,A10,A11,A12,YELLOW_0:19;
  end;
  hence thesis;
end;
