reserve Y for non empty set;
reserve Y for non empty set;
reserve Y for non empty set;
reserve Y for non empty set,
  a,b,c,d,e,f,g for Function of Y,BOOLEAN;
reserve Y for non empty set;

theorem
  for a1,a2,b1,b2,b3 being Function of Y,BOOLEAN holds (a1 '&' a2)
'or' (b1 '&' b2 '&' b3)= (a1 'or' b1) '&' (a1 'or' b2) '&' (a1 'or' b3) '&' (a2
  'or' b1) '&' (a2 'or' b2) '&' (a2 'or' b3)
proof
  let a1,a2,b1,b2,b3 be Function of Y,BOOLEAN;
  (a1 'or' b1) '&' (a1 'or' b2) '&' (a1 'or' b3) '&' (a2 'or' b1) '&' (a2
'or' b2) '&' (a2 'or' b3) =(a1 'or' (b1 '&' b2)) '&' (a1 'or' b3) '&' (a2 'or'
  b1) '&' (a2 'or' b2) '&' (a2 'or' b3) by BVFUNC_1:11
    .=(a1 'or' (b1 '&' b2 '&' b3)) '&' (a2 'or' b1) '&' (a2 'or' b2) '&' (a2
  'or' b3) by BVFUNC_1:11
    .=(a1 'or' (b1 '&' b2 '&' b3)) '&' ((a2 'or' b1) '&' (a2 'or' b2)) '&' (
  a2 'or' b3) by BVFUNC_1:4
    .=(a1 'or' (b1 '&' b2 '&' b3)) '&' (((a2 'or' b1) '&' (a2 'or' b2)) '&'
  (a2 'or' b3)) by BVFUNC_1:4
    .=(a1 'or' (b1 '&' b2 '&' b3)) '&' ((a2 'or' (b1 '&' b2)) '&' (a2 'or'
  b3)) by BVFUNC_1:11
    .=(a1 'or' (b1 '&' b2 '&' b3)) '&' (a2 'or' (b1 '&' b2 '&' b3)) by
BVFUNC_1:11
    .=(a1 '&' a2) 'or' (b1 '&' b2 '&' b3) by BVFUNC_1:11;
  hence thesis;
end;
