reserve A, B, X, Y for set;

theorem Th24:
  for R being upper-bounded Semilattice, X being Subset of [:R,R:]
  st ex_inf_of (inf_op R).:X,R holds inf_op R preserves_inf_of X
proof
  let R be upper-bounded Semilattice;
  set f = inf_op R;
  let X be Subset of [:R,R:] such that
A1: ex_inf_of f.:X,R and
A2: ex_inf_of X,[:R,R:];
  thus ex_inf_of f.:X,R by A1;
A3: dom f = the carrier of [:R,R:] by FUNCT_2:def 1;
  then
A4: dom f = [:the carrier of R,the carrier of R:] by YELLOW_3:def 2;
A5: for b being Element of R st b is_<=_than f.:X holds f.inf X >= b
  proof
    let b be Element of R such that
A6: b is_<=_than f.:X;
    X is_>=_than [b,b]
    proof
      let c be Element of [:R,R:];
      assume c in X;
      then f.c in f.:X by A3,FUNCT_1:def 6;
      then
A7:   b <= f.c by A6;
      consider s, t being object such that
A8:   s in the carrier of R & t in the carrier of R and
A9:   c = [s,t] by A3,A4,ZFMISC_1:def 2;
      reconsider s, t as Element of R by A8;
A10:  f.c = f.(s,t) by A9
        .= s "/\" t by WAYBEL_2:def 4;
      s "/\" t <= t by YELLOW_0:23;
      then
A11:  b <= t by A7,A10,ORDERS_2:3;
      s "/\" t <= s by YELLOW_0:23;
      then b <= s by A7,A10,ORDERS_2:3;
      hence [b,b] <= c by A9,A11,YELLOW_3:11;
    end;
    then [b,b] <= inf X by A2,YELLOW_0:def 10;
    then f.(b,b) <= f.inf X by WAYBEL_1:def 2;
    then b "/\" b <= f.inf X by WAYBEL_2:def 4;
    hence b <= f.inf X by YELLOW_0:25;
  end;
  inf X is_<=_than X by A2,YELLOW_0:def 10;
  then f.inf X is_<=_than f.:X by YELLOW_2:13;
  hence thesis by A1,A5,YELLOW_0:def 10;
end;
