reserve X for set,
  x,y,z for Element of BooleLatt X,
  s for set;
reserve y for Element of BooleLatt X;
reserve L for Lattice,
  p,q for Element of L;
reserve A for RelStr,
  a,b,c for Element of A;
reserve A for non empty RelStr,
  a,b,c,c9 for Element of A;
reserve V for with_suprema antisymmetric RelStr,
  u1,u2,u3,u4 for Element of V;
reserve N for with_infima antisymmetric RelStr,
  n1,n2,n3,n4 for Element of N;
reserve K for with_suprema with_infima reflexive antisymmetric RelStr,
  k1,k2,k3 for Element of K;

theorem Th26:
  BooleLatt X is \/-distributive
proof
  let x be set;
  set B = BooleLatt X;
  let a,b,c be Element of B such that
A1: x is_less_than a and
A2: for d being Element of B st x is_less_than d holds a [= d and
A3: {b"/\" a9 where a9 is Element of B: a9 in x} is_less_than c and
A4: for d being Element of B st
  {b"/\"a9 where a9 is Element of B: a9 in x} is_less_than d holds c [= d;
  set Y = {b"/\"a9 where a9 is Element of B: a9 in x};
A5: carr(B) = bool X by Def1;
A6: Y c= bool X
  proof
    let z be object;
    assume z in Y;
    then ex a9 being Element of B st z = b"/\"a9 & a9 in x;
    hence thesis by A5;
  end;
A7: (union Y) c= union bool X by A6,ZFMISC_1:77;
  union bool X = X by ZFMISC_1:81;
  then reconsider
  p = union (x /\ bool X),q = union Y as Element of B by A7,Def1;
  now
    let y be set;
    assume
A8: y in x /\ bool X;
    then
A9: y in x by XBOOLE_0:def 4;
    reconsider y9 = y as Element of B by A5,A8;
    y9 [= a by A1,A9;
    hence y c= a by Th2;
  end;
  then
A10: p c= a by ZFMISC_1:76;
A11: x is_less_than p
  proof
    let q be Element of B;
    assume q in x;
    then q in x /\ bool X by A5,XBOOLE_0:def 4;
    then q c= p by ZFMISC_1:74;
    hence q [= p by Th2;
  end;
A12: p [= a by A10,Th2;
  a [= p by A2,A11;
  then
A13: a = p by A12;
  now
    let y be set;
    assume
A14: y in Y;
    then consider a9 being Element of B such that
A15: y = b"/\"a9 and a9 in x;
    b"/\"a9 [= c by A3,A14,A15;
    hence y c= c by A15,Th2;
  end;
  then
A16: q c= c by ZFMISC_1:76;
A17: Y is_less_than q
  by ZFMISC_1:74,Th2;
A18: q [= c by A16,Th2;
  c [= q by A4,A17;
  then
A19: c = q by A18;
  b /\ a c= c
  proof
    let z be object;
    assume
A20: z in b /\ a;
    then
A21: z in b by XBOOLE_0:def 4;
    z in a by A20,XBOOLE_0:def 4;
    then consider y being set such that
A22: z in y and
A23: y in x /\ bool X by A13,TARSKI:def 4;
A24: y in x by A23,XBOOLE_0:def 4;
    reconsider y as Element of B by A5,A23;
A25: b"/\"y in Y by A24;
    z in b /\ y by A21,A22,XBOOLE_0:def 4;
    hence thesis by A19,A25,TARSKI:def 4;
  end;
  hence b"/\"a [= c by Th2;
end;
