
theorem Th46:
  for L being non empty RelStr, X,Y being set st for x being
  Element of L holds x is_>=_than X iff x is_>=_than Y holds ex_sup_of X,L
  implies ex_sup_of Y,L
proof
  let L be non empty RelStr, X,Y be set such that
A1: for x being Element of L holds x is_>=_than X iff x is_>=_than Y;
  given a being Element of L such that
A2: X is_<=_than a and
A3: for b being Element of L st X is_<=_than b holds a <= b and
A4: for c being Element of L st X is_<=_than c & for b being Element of
  L st X is_<=_than b holds c <= b holds c = a;
  take a;
  thus Y is_<=_than a by A1,A2;
  thus for b being Element of L st Y is_<=_than b holds a <= b by A1,A3;
  let c be Element of L;
  assume
A5: Y is_<=_than c;
  assume
A6: for b being Element of L st Y is_<=_than b holds c <= b;
A7: for b being Element of L st X is_<=_than b holds c <= b by A1,A6;
  X is_<=_than c by A1,A5;
  hence thesis by A4,A7;
end;
