
theorem Th23:
  for L being with_suprema Poset, x,y being Element of L holds x
  "\/"y = (x~)"/\"(y~)
proof
  let L be with_suprema Poset, x,y be Element of L;
  x"\/"y >= y by YELLOW_0:22;
  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_0:22;
    (~d)~ = ~d;
    hence (x"\/"y)~ >= d by A4,LATTICE3:9;
  end;
  x"\/"y >= x by YELLOW_0:22;
  then (x"\/"y)~ <= x~ by LATTICE3:9;
  hence thesis by A1,A3,YELLOW_0:23;
end;
