
theorem Th11:
  for N being complete non empty Poset, x being Element of N, X
  being non empty Subset of N holds x"/\" preserves_inf_of X
proof
  let N be complete non empty Poset, x be Element of N, X be non empty
  Subset of N such that
A1: ex_inf_of X,N;
A2: for b being Element of N st b is_<=_than (x"/\").:X holds x"/\".inf X >= b
  proof
    consider y being object such that
A3: y in X by XBOOLE_0:def 1;
    reconsider y as Element of N by A3;
    let b be Element of N such that
A4: b is_<=_than (x"/\").:X;
A5: (x"/\").: X = {x"/\"z where z is Element of N: z in X} by WAYBEL_1:61;
    then x "/\" y in (x"/\").:X by A3;
    then
A6: b <= x "/\" y by A4;
    X is_>=_than b
    proof
      let c be Element of N;
      assume c in X;
      then x "/\" c in (x"/\").:X by A5;
      then
A7:   b <= x "/\" c by A4;
      x "/\" c <= c by YELLOW_0:23;
      hence b <= c by A7,ORDERS_2:3;
    end;
    then
A8: b <= inf X by A1,YELLOW_0:def 10;
    x "/\" y <= x by YELLOW_0:23;
    then b <= x by A6,ORDERS_2:3;
    then b "/\" b <= x "/\" inf X by A8,YELLOW_3:2;
    then b <= x "/\" inf X by YELLOW_0:25;
    hence b <= x"/\".inf X by WAYBEL_1:def 18;
  end;
  thus ex_inf_of (x"/\").:X,N by YELLOW_0:17;
  inf X is_<=_than X by A1,YELLOW_0:def 10;
  then x"/\".inf X is_<=_than (x"/\").:X by YELLOW_2:13;
  hence thesis by A2,YELLOW_0:33;
end;
