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

theorem Th43:
  for L being complete LATTICE, x being Element of L holds
  meet the set of all  (DownMap I).x where I is Ideal of L  = waybelow x
proof
  let L be complete LATTICE, x be Element of L;
  set A = the set of all  (DownMap I).x where I is Ideal of L ;
  set A1 = { (DownMap I).x where I is Ideal of L : x <= sup I};
A1: A1 c= { (downarrow x) /\ I where I is Ideal of L : x <= sup I}
  proof
    let a be object;
    assume a in A1;
    then consider I1 be Ideal of L such that
A2: a = (DownMap I1).x and
A3: x <= sup I1;
    a = (downarrow x) /\ I1 by A2,A3,Def16;
    hence thesis by A3;
  end;
  { (downarrow x) /\ I where I is Ideal of L : x <= sup I} c= A1
  proof
    let a be object;
    assume a in { (downarrow x) /\ I where I is Ideal of L : x <= sup I};
    then consider I1 be Ideal of L such that
A4: a = (downarrow x) /\ I1 and
A5: x <= sup I1;
    a = (DownMap I1).x by A4,A5,Def16;
    hence thesis by A5;
  end;
  then
A6: A1 = { (downarrow x) /\ I where I is Ideal of L : x <= sup I} by A1;
  set A2 = { (DownMap I).x where I is Ideal of L : not x <= sup I};
A7: A2 c= { downarrow x where I is Ideal of L : not x <= sup I}
  proof
    let a be object;
    assume a in A2;
    then
A8: ex I1 be Ideal of L st ( a = (DownMap I1).x)&( not x <= sup I1);
    then a = (downarrow x) by Def16;
    hence thesis by A8;
  end;
  { (downarrow x) where I is Ideal of L : not x <= sup I} c= A2
  proof
    let a be object;
    assume a in { (downarrow x) where I is Ideal of L : not x <= sup I};
    then consider I1 be Ideal of L such that
A9: a = (downarrow x) and
A10: not x <= sup I1;
    a = (DownMap I1).x by A9,A10,Def16;
    hence thesis by A10;
  end;
  then
A11: A2 = { downarrow x where I is Ideal of L : not x <= sup I} by A7;
A12: A c= A1 \/ A2
  proof
    let a be object;
    assume a in A;
    then consider I2 be Ideal of L such that
A13: a = (DownMap I2).x;
    x <= sup I2 or not x <= sup I2;
    then a in A1 or a in A2 by A13;
    hence thesis by XBOOLE_0:def 3;
  end;
  A1 \/ A2 c= A
  proof
    let a be object;
    assume a in A1 \/ A2;
    then a in A1 or a in A2 by XBOOLE_0:def 3;
    then consider I2,I3 be Ideal of L such that
A14: a = (DownMap I2).x & x <= sup I2 or a = (DownMap I3).x & not x <= sup I3;
    per cases by A14;
    suppose a = (DownMap I2).x & x <= sup I2;
      hence thesis;
    end;
    suppose a = (DownMap I3).x & not x <= sup I3;
      hence thesis;
    end;
  end;
  then
A15: A = A1 \/ A2 by A12;
  per cases;
  suppose
A16: x = Bottom L;
A17: A2 = {}
    proof
      assume A2 <> {};
      then reconsider A29 = A2 as non empty set;
      set a = the Element of A29;
      a in A29;
      then ex I1 be Ideal of L st ( a = downarrow x)&( not x <= sup I1
      ) by A11;
      hence contradiction by A16,YELLOW_0:44;
    end;
    set Dx = downarrow x;
    x <= sup Dx by A16,YELLOW_0:44;
    then
A18: Dx /\ Dx in A1 by A6;
    A1 c= { K where K is Ideal of L : x <= sup K}
    proof
      let a be object;
      assume a in A1;
      then ex H be Ideal of L st ( a = (downarrow x) /\ H)&( x <= sup H) by A6;
      then reconsider a9 = a as Ideal of L by YELLOW_2:40;
      x <= sup a9 by A16,YELLOW_0:44;
      hence thesis;
    end;
    then
A19: meet { K where K is Ideal of L : x <= sup K} c= meet A1 by A18,SETFAM_1:6;
A20: meet A1 = {x}
    proof
      reconsider I9 = downarrow x as Ideal of L;
      x <= sup I9 by A16,YELLOW_0:44;
      then (downarrow x) /\ I9 in A1 by A6;
      then {x} in A1 by A16,Th23;
      hence meet A1 c= {x} by SETFAM_1:3;
      for Z1 be set st Z1 in A1 holds {x} c= Z1
      proof
        let Z1 be set;
        assume Z1 in A1;
        then consider Z19 be Ideal of L such that
A21:    Z1 = (downarrow x) /\ Z19 and x <= sup Z19 by A6;
A22:    {x} c= Z19 by A16,Lm4;
        {x} c= downarrow x by A16,Th23;
        hence thesis by A21,A22,XBOOLE_1:19;
      end;
      hence thesis by A18,SETFAM_1:5;
    end;
    set E = the Ideal of L;
    x <= sup E by A16,YELLOW_0:44;
    then
A23: E in { K where K is Ideal of L : x <= sup K};
    for Z1 be set st Z1 in { K where K is Ideal of L : x <= sup K}
    holds meet A1 c= Z1
    proof
      let Z1 be set;
      assume Z1 in { K where K is Ideal of L : x <= sup K};
      then ex K1 be Ideal of L st ( K1 = Z1)&( x <= sup K1);
      hence thesis by A16,A20,Lm4;
    end;
    then meet A1 c= meet { K where K is Ideal of L : x <= sup K} by A23,
SETFAM_1:5;
    then meet A1 = meet { K where K is Ideal of L : x <= sup K} by A19;
    hence thesis by A15,A17,Th34;
  end;
  suppose
A24: Bottom L <> x;
    set O = downarrow Bottom L;
A25: sup O = Bottom L by WAYBEL_0:34;
    then not x < sup O by ORDERS_2:6,YELLOW_0:44;
    then not x <= sup O by A24,A25,ORDERS_2:def 6;
    then
A26: downarrow x in A2 by A11;
    reconsider O1 = downarrow x as Ideal of L;
A27: x <= sup O1 by WAYBEL_0:34;
    downarrow x = downarrow x /\ O1;
    then
A28: downarrow x in A1 by A6,A27;
A29: meet A2 c= downarrow x by A26,SETFAM_1:3;
    now
      let Z1 be set;
      assume Z1 in A2;
      then ex L1 be Ideal of L st ( Z1 = downarrow x)&( not x <= sup
      L1) by A11;
      hence downarrow x c= Z1;
    end;
    then
    downarrow x c= meet A2 by A26,SETFAM_1:5;
    then
A30: meet A2 = downarrow x by A29;
A31: meet A = (meet A1) /\ (meet A2) by A15,A26,A28,SETFAM_1:9;
A32: meet A1 c= downarrow x by A28,SETFAM_1:3;
A33: meet A = meet A1 by A28,A30,A31,SETFAM_1:3,XBOOLE_1:28;
A34: meet A1 c= (downarrow x) /\ meet { I where I is Ideal of L : x <= sup I}
    proof
      let a be object;
      assume
A35:  a in meet A1;
      thus
      a in (downarrow x) /\ meet { I where I is Ideal of L : x <= sup I}
      proof
        reconsider L9 = [#]L as Subset of L;
        ex_sup_of L9,L by YELLOW_0:17;
        then L9 is_<=_than sup L9 by YELLOW_0:def 9;
        then x <= sup L9;
        then
A36:    L9 in { I where I is Ideal of L : x <= sup I};
        now
          let Y1 be set;
          assume Y1 in { I where I is Ideal of L : x <= sup I};
          then consider Y19 be Ideal of L such that
A37:      Y1 = Y19 and
A38:      x <= sup Y19;
A39:      (downarrow x) /\ Y19 c= Y19 by XBOOLE_1:17;
          (downarrow x) /\ Y19 in A1 by A6,A38;
          then a in (downarrow x) /\ Y19 by A35,SETFAM_1:def 1;
          hence a in Y1 by A37,A39;
        end;
        then a in meet { I where I is Ideal of L : x <= sup I} by A36,
SETFAM_1:def 1;
        hence thesis by A32,A35,XBOOLE_0:def 4;
      end;
    end;
    (downarrow x) /\ meet { I where I is Ideal of L : x <= sup I} c= meet A1
    proof
      let a be object;
      assume
A40:  a in (downarrow x) /\ meet { I where I is Ideal of L : x <= sup I};
      then
A41:  a in downarrow x by XBOOLE_0:def 4;
A42:  a in meet { I where I is Ideal of L : x <= sup I} by A40,XBOOLE_0:def 4;
      now
        let Y1 be set;
        assume Y1 in { (downarrow x) /\ I where I is Ideal of L : x <= sup I};
        then consider I1 be Ideal of L such that
A43:    Y1 = (downarrow x) /\ I1 and
A44:    x <= sup I1;
        I1 in { I where I is Ideal of L : x <= sup I} by A44;
        then a in I1 by A42,SETFAM_1:def 1;
        hence a in Y1 by A41,A43,XBOOLE_0:def 4;
      end;
      hence a in meet A1 by A6,A28,SETFAM_1:def 1;
    end;
    then (downarrow x) /\ meet { I where I is Ideal of L : x <= sup I} = meet
    A1 by A34;
    then (downarrow x) /\ waybelow x = meet A1 by Th34;
    hence thesis by A33,WAYBEL_3:11,XBOOLE_1:28;
  end;
end;
