reserve a for set;
reserve L for lower-bounded sup-Semilattice;
reserve x for Element of L;

theorem Th36:
  for L being with_infima antisymmetric reflexive RelStr, x being Element of L
  for D being non empty lower Subset of L holds
  {x} "/\" D = (downarrow x) /\ D
proof
  let L be with_infima antisymmetric reflexive RelStr, x be Element of L;
  let D be non empty lower Subset of L;
  A1: {
x} "/\" D = { x9 "/\" y where x9, y is Element of L : x9 in {x} & y in D }
  by YELLOW_4:def 4;
  thus {x} "/\" D c= (downarrow x) /\ D
  proof
    let a be object;
    assume a in {x} "/\" D;
    then a in { x9 "/\" y where x9, y is Element of L : x9 in {x} & y in D }
    by YELLOW_4:def 4;
    then consider x9, y be Element of L such that
A2: a = x9 "/\" y and
A3: x9 in {x} and
A4: y in D;
    reconsider a9 = a as Element of L by A2;
A5: x9 = x by A3,TARSKI:def 1;
A6: ex v being Element of L st x9 >= v & y >= v &
    for c being Element of L st x9 >= c & y >= c holds v >= c
    by LATTICE3:def 11;
    then
A7: x9 "/\" y <= x9 by LATTICE3:def 14;
A8: x9 "/\" y <= y by A6,LATTICE3:def 14;
A9: a in downarrow x by A2,A5,A7,WAYBEL_0:17;
    a9 in D by A2,A4,A8,WAYBEL_0:def 19;
    hence thesis by A9,XBOOLE_0:def 4;
  end;
  thus (downarrow x) /\ D c= {x} "/\" D
  proof
    let a be object;
    assume
A10: a in (downarrow x) /\ D;
    then
A11: a in downarrow x by XBOOLE_0:def 4;
    reconsider a9 = a as Element of D by A10,XBOOLE_0:def 4;
A12: x in {x} by TARSKI:def 1;
    a9 <= x by A11,WAYBEL_0:17;
    then a9 = a9 "/\" x by YELLOW_0:25;
    hence thesis by A1,A12;
  end;
end;
