
theorem Th59:
  for S being non empty RelStr, x,t being Element of S holds {s
  where s is Element of S: x"/\"s <= t} = (x "/\")"(downarrow t)
proof
  let S be non empty RelStr, x,t be Element of S;
  hereby
    let a be object;
    assume a in {s where s is Element of S: x"/\"s <= t};
    then consider s being Element of S such that
A1: a = s and
A2: x"/\"s <= t;
    (x "/\").s <= t by A2,Def18;
    then (x"/\").s in downarrow t by WAYBEL_0:17;
    hence a in (x "/\")"(downarrow t) by A1,FUNCT_2:38;
  end;
  let s be object;
  assume
A3: s in (x "/\")"(downarrow t);
  then reconsider s as Element of S;
  (x "/\").s in downarrow t by A3,FUNCT_2:38;
  then x"/\"s in downarrow t by Def18;
  then x"/\"s <= t by WAYBEL_0:17;
  hence thesis;
end;
