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 Th27:
  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_greater_than a and
A2: for d being Element of B st x is_greater_than d holds d [= a and
A3: {b"\/"a9 where a9 is Element of B: a9 in x} is_greater_than c and
A4: for d being Element of B st
  {b"\/"a9 where a9 is Element of B: a9 in x} is_greater_than d holds d [= c;
  set x9 = {e` where e is Element of B: e in x},
  y = {b"\/"e where e is Element of B: e in x},
  y9 = {e` where e is Element of B: e in y},
  z = {b`"/\"e where e is Element of B: e in x9};
A5: z = y9
  proof
    thus z c= y9
    proof
      let s be object;
      assume s in z;
      then consider e being Element of B such that
A6:   s = b`"/\"e and
A7:   e in x9;
      consider i being Element of B such that
A8:   e = i` and
A9:   i in x by A7;
A10:  b"\/"i in y by A9;
      (b"\/"i)` = s by A6,A8,LATTICES:24;
      hence thesis by A10;
    end;
    let s be object;
    assume s in y9;
    then consider e being Element of B such that
A11: s = e` and
A12: e in y;
    consider i being Element of B such that
A13: e = b"\/"i and
A14: i in x by A12;
A15: i` in x9 by A14;
    s = b`"/\"i` by A11,A13,LATTICES:24;
    hence thesis by A15;
  end;
A16: c`` = c;
A17: x9 is_less_than a` by A1,Th24;
A18: for d being Element of B st x9 is_less_than d holds a` [= d
  proof
    let d be Element of B;
A19: d`` = d;
    assume x9 is_less_than d;
    then x is_greater_than d` by A19,Th24;
    hence thesis by A2,A19,LATTICES:26;
  end;
A20: z is_less_than c` by A3,A5,Th24;
A21: for d being Element of B st z is_less_than d holds c` [= d
  proof
    let d be Element of B;
A22: d`` = d;
    assume z is_less_than d;
    then y is_greater_than d` by A5,A22,Th24;
    hence thesis by A4,A22,LATTICES:26;
  end;
  B is \/-distributive by Th26;
  then
A23: b`"/\"a` [= c` by A17,A18,A20,A21;
  (b`"/\"a`)` = b``"\/" a`` by LATTICES:23;
  hence c [= b"\/"a by A16,A23,LATTICES:26;
end;
