
theorem Th47:
  for L being non empty RelStr, X,Y being set st ex_sup_of X,L &
for x being Element of L holds x is_>=_than X iff x is_>=_than Y holds "\/"(X,L
  ) = "\/"(Y,L)
proof
  let L be non empty RelStr, X,Y be set;
  assume
A1: ex_sup_of X,L;
  assume
A2: for x being Element of L holds x is_>=_than X iff x is_>=_than Y;
A3: now
    let b be Element of L;
    assume Y is_<=_than b;
    then X is_<=_than b by A2;
    hence "\/"(X,L) <= b by A1,Def9;
  end;
  X is_<=_than "\/"(X,L) by A1,Def9;
  then
A4: Y is_<=_than "\/"(X,L) by A2;
  ex_sup_of Y,L by A1,A2,Th46;
  hence thesis by A4,A3,Def9;
end;
