reserve L for Boolean non empty RelStr;
reserve a,b,c,d for Element of L;

theorem Th36:
  'not' (a"\/"b) = 'not' a"/\" 'not' b & 'not' (a"/\"b) = 'not' a "\/" 'not' b
proof
A1: (a"\/"b) "/\" ('not' a"/\" 'not' b) = ((a"\/"b) "/\" 'not' a)"/\" 'not'
  b by LATTICE3:16
    .= (('not' a"/\"a)"\/"('not' a"/\"b))"/\" 'not' b by WAYBEL_1:def 3
    .= (Bottom L "\/"('not' a"/\"b))"/\" 'not' b by Th34
    .= ('not' a"/\"b)"/\" 'not' b by WAYBEL_1:3
    .= 'not' a"/\"(b"/\" 'not' b) by LATTICE3:16
    .= 'not' a"/\" Bottom L by Th34
    .= Bottom L by WAYBEL_1:3;
  (a"\/"b) "\/" ('not' a"/\" 'not' b) = a"\/"(b "\/" ('not' a"/\" 'not' b)
  ) by LATTICE3:14
    .= a"\/"((b"\/"'not' a)"/\"(b"\/"'not' b)) by Th17
    .= a"\/"((b"\/"'not' a)"/\" Top L) by Th34
    .= a"\/"(b"\/"'not' a) by WAYBEL_1:4
    .= (a"\/"'not' a)"\/"b by LATTICE3:14
    .= Top L"\/"b by Th34
    .= Top L by WAYBEL_1:4;
  then ('not' a"/\"'not' b) is_a_complement_of (a"\/" b) by A1,WAYBEL_1:def 23;
  hence 'not' (a"\/"b) = 'not' a"/\" 'not' b by Th11;
A2: (a"/\"b) "/\" ('not' a"\/" 'not' b) = a"/\"(b"/\" ('not' a"\/" 'not' b))
  by LATTICE3:16
    .= a"/\"((b"/\"'not' a)"\/"(b"/\"'not' b)) by WAYBEL_1:def 3
    .= a"/\"((b"/\"'not' a)"\/" Bottom L) by Th34
    .= a"/\"(b"/\"'not' a) by WAYBEL_1:3
    .= (a"/\"'not' a)"/\"b by LATTICE3:16
    .= Bottom L"/\"b by Th34
    .= Bottom L by WAYBEL_1:3;
  (a"/\"b) "\/" ('not' a"\/" 'not' b) = ((a"/\"b) "\/"'not' a)"\/" 'not' b
  by LATTICE3:14
    .= (('not' a"\/"a) "/\" ('not' a "\/"b) )"\/" 'not' b by Th17
    .= (Top L "/\" ('not' a "\/"b) )"\/" 'not' b by Th34
    .= ('not' a "\/"b)"\/" 'not' b by WAYBEL_1:4
    .= 'not' a"\/"(b"\/" 'not' b) by LATTICE3:14
    .= 'not' a"\/"Top L by Th34
    .= Top L by WAYBEL_1:4;
  then ('not' a"\/" 'not' b) is_a_complement_of (a"/\" b) by A2,WAYBEL_1:def 23
;
  hence 'not' (a"/\"b) = 'not' a"\/" 'not' b by Th11;
end;
