
theorem Th21:
  for L being with_infima Poset, x,y being Element of L holds x
  "/\"y = (x~)"\/"(y~)
proof
  let L be with_infima Poset, x,y be Element of L;
  x"/\"y <= y by YELLOW_0:23;
  then
A1: (x"/\"y)~ >= y~ by LATTICE3:9;
A2: ~(x~) = x~ & ~(y~) = y~;
A3: now
    let d be Element of L opp;
    assume d >= x~ & d >= y~;
    then ~d <= x & ~d <= y by A2,Th1;
    then
A4: ~d <= x"/\"y by YELLOW_0:23;
    (~d)~ = ~d;
    hence (x"/\"y)~ <= d by A4,LATTICE3:9;
  end;
  x"/\"y <= x by YELLOW_0:23;
  then (x"/\"y)~ >= x~ by LATTICE3:9;
  hence thesis by A1,A3,YELLOW_0:22;
end;
