
theorem Th13:
  for L being non empty RelStr, X being set st ex_inf_of X,L or
  ex_sup_of X,L opp holds "/\"(X,L) = "\/"(X,L opp)
proof
  let L be non empty RelStr, X be set;
  assume
A1: ex_inf_of X,L or ex_sup_of X,L opp;
  then
A2: ex_inf_of X,L by Th11;
  then "/\"(X,L) is_<=_than X by YELLOW_0:def 10;
  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 10;
    hence x >= "/\"(X,L)~ by Th2;
  end;
  ex_sup_of X,L opp by A1,Th11;
  hence thesis by A3,A4,YELLOW_0:def 9;
end;
