
theorem Th40:
for S being non empty RelStr, T being non empty reflexive antisymmetric RelStr
  for t being Element of T for X being non empty Subset of S
  holds S --> t preserves_sup_of X & S --> t preserves_inf_of X
proof
  let S be non empty RelStr;
  let T be non empty reflexive antisymmetric RelStr;
  let t be Element of T;
  let X be non empty Subset of S;
  set f = S --> t;
A1: f.:X = {t}
  proof
    thus f.:X c= {t} by FUNCOP_1:81;
    set x = the Element of X;
    f.x = t by FUNCOP_1:7;
    then t in f.:X by FUNCT_2:35;
    hence thesis by ZFMISC_1:31;
  end;
A2: f.sup X = t by FUNCOP_1:7;
A3: f.inf X = t by FUNCOP_1:7;
A4: inf {t} = t by YELLOW_0:39;
A5: sup {t} = t by YELLOW_0:39;
A6: ex_sup_of {t}, T by YELLOW_0:38;
  ex_inf_of {t}, T by YELLOW_0:38;
  hence thesis by A1,A2,A3,A4,A5,A6;
end;
