
theorem Th31:
  for L being non empty transitive reflexive RelStr, X be Subset of L
  for x be Element of L holds x is_>=_than X iff x is_>=_than downarrow X
proof
  let L be non empty transitive reflexive RelStr, X be Subset of L;
  let x be Element of L;
  thus x is_>=_than X implies x is_>=_than downarrow X
  proof
    assume
A1: for y being Element of L st y in X holds x >= y;
    let y be Element of L;
    assume y in downarrow X;
    then consider z being Element of L such that
A2: y <= z and
A3: z in X by Def15;
    x >= z by A1,A3;
    hence thesis by A2,ORDERS_2:3;
  end;
  assume
A4: for y being Element of L st y in downarrow X holds x >= y;
  let y be Element of L;
  assume
A5: y in X;
  y <= y;
  then y in downarrow X by A5,Def15;
  hence thesis by A4;
end;
