reserve A, B, X, Y for set;

theorem
  for N being Semilattice, A being Subset of N st subrelstr A is
  meet-inheriting holds A is filtered
proof
  let N be Semilattice, A be Subset of N such that
A1: subrelstr A is meet-inheriting;
  let x, y be Element of N such that
A2: x in A & y in A;
  take x"/\"y;
A3: the carrier of subrelstr A = A by YELLOW_0:def 15;
  ex_inf_of {x,y},N by YELLOW_0:21;
  then inf {x,y} in the carrier of subrelstr A by A1,A2,A3;
  hence x"/\"y in A by A3,YELLOW_0:40;
  thus x"/\"y <= x & x"/\"y <= y by YELLOW_0:23;
end;
