
theorem Th49:
  for L being non empty RelStr, X,Y being set st ex_inf_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_inf_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,Def10;
  end;
  X is_>=_than "/\"(X,L) by A1,Def10;
  then
A4: Y is_>=_than "/\"(X,L) by A2;
  ex_inf_of Y,L by A1,A2,Th48;
  hence thesis by A4,A3,Def10;
end;
