
theorem Th12:
  for L being non empty RelStr, X being set st ex_sup_of X,L or
  ex_inf_of X,L opp holds "\/"(X,L) = "/\"(X,L opp)
proof
  let L be non empty RelStr, X be set;
  assume
A1: ex_sup_of X,L or ex_inf_of X,L opp;
  then
A2: ex_sup_of X,L by Th10;
  then "\/"(X,L) is_>=_than X by YELLOW_0:def 9;
  then
A3: "\/"(X,L)~ is_<=_than X by Th8;
A4: now
    let x be Element of L opp;
    assume x is_<=_than X;
    then ~x is_>=_than X by Th9;
    then ~x >= "\/"(X,L) by A2,YELLOW_0:def 9;
    hence x <= "\/"(X,L)~ by Th2;
  end;
  ex_inf_of X,L opp by A1,Th10;
  hence thesis by A3,A4,YELLOW_0:def 10;
end;
