
theorem Th25:
  for L be complete Boolean LATTICE for x be Element of L for A be
  Subset of L st A c= ATOM L holds x in A iff x is atom & x <= sup A
proof
  let L be complete Boolean LATTICE;
  let x be Element of L;
  let A be Subset of L;
  assume
A1: A c= ATOM L;
  thus x in A implies x is atom & x <= sup A
  proof
    assume
A2: x in A;
    hence x is atom by A1,Def2;
    sup A is_>=_than A by YELLOW_0:32;
    hence thesis by A2,LATTICE3:def 9;
  end;
  thus x is atom & x <= sup A implies x in A
  proof
    assume that
A3: x is atom and
A4: x <= sup A and
A5: not x in A;
    now
      let b be Element of L;
      assume b in { x "/\" y where y is Element of L: y in A };
      then consider y be Element of L such that
A6:   b = x "/\" y and
A7:   y in A;
      y is atom by A1,A7,Def2;
      hence b <= Bottom L by A3,A5,A6,A7,Th24;
    end;
    then
A8: Bottom L is_>=_than { x "/\" y where y is Element of L: y in A } by
LATTICE3:def 9;
    x = x "/\" sup A by A4,YELLOW_0:25
      .= "\/"({ x "/\" y where y is Element of L: y in A },L) by Th23;
    then x <= Bottom L by A8,YELLOW_0:32;
    then x = Bottom L by YELLOW_5:19;
    hence contradiction by A3;
  end;
end;
