
theorem Th21:
  for L being Semilattice, F being upper filtered Subset of L
  holds F "/\" F = F
proof
  let L be Semilattice, F be upper filtered Subset of L;
  thus F "/\" F c= F
  proof
    let x be object;
    assume x in F "/\" F;
    then consider y, z being Element of L such that
A1: x = y "/\" z and
A2: y in F & z in F;
    consider t being Element of L such that
A3: t in F and
A4: t <= y & t <= z by A2,WAYBEL_0:def 2;
    y "/\" z >= t by A4,YELLOW_0:23;
    hence thesis by A1,A3,WAYBEL_0:def 20;
  end;
  thus thesis by YELLOW_4:40;
end;
