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;

theorem Th23:
  for a1,b1,c1,a2,b2,c2 being Function of Y,BOOLEAN holds (
  b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2 '&' b2)
  '&' 'not'( a2 '&' c2) '<' (a2 'imp' a1)
proof
  let a1,b1,c1,a2,b2,c2 be Function of Y,BOOLEAN;
A1: ((b1 'or' c1) 'imp' (b2 'or' c2)) '&' ((b2 'or' c2) 'imp' 'not' a2) '&'
((b1 'or' c1) 'imp' 'not' a2) 'imp' ((b1 'or' c1) 'imp' 'not' a2) = I_el(Y) by
Th38;
A2: (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2
  '&' b2) '&' 'not'( a2 '&' c2) 'imp' (c1 'imp' c2) = I_el(Y)
     by Lm4,BVFUNC_1:16;
A3: ((b1 'imp' b2) '&' (c1 'imp' c2)) 'imp' ((b1 'or' c1) 'imp' (b2 'or' c2
  )) = I_el(Y) by Th21,BVFUNC_1:16;
  (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2
  '&' b2) '&' 'not'( a2 '&' c2) 'imp' (b1 'imp' b2) = I_el(Y)
          by Lm4,BVFUNC_1:16;
  then (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2
  '&' b2) '&' 'not'( a2 '&' c2) 'imp' (b1 'imp' b2) '&' (c1 'imp' c2) = I_el(Y)
  by A2,th18;
  then
A4: (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2
'&' b2) '&' 'not'( a2 '&' c2) 'imp' ((b1 'or' c1) 'imp' (b2 'or' c2)) = I_el(Y)
  by A3,BVFUNC_5:9;
A5: (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2
  '&' b2) '&' 'not'( a2 '&' c2) 'imp' 'not'( a2 '&' c2) = I_el(Y) by Lm1,
BVFUNC_1:16;
  (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2
  '&' b2) '&' 'not'( a2 '&' c2) 'imp' 'not'( a2 '&' b2) = I_el(Y) by Lm2,
BVFUNC_1:16;
  then (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2
'&' b2) '&' 'not'( a2 '&' c2) 'imp' ('not'( a2 '&' b2) '&' 'not'( a2 '&' c2)) =
  I_el(Y) by A5,th18;
  then
A6: (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2
  '&' b2) '&' 'not'( a2 '&' c2) 'imp' ((b1 'or' c1) 'imp' (b2 'or' c2)) '&' (
  'not'( a2 '&' b2) '&' 'not' (a2 '&' c2)) = I_el(Y) by A4,th18;
  'not'( a2 '&' b2) '&' 'not'( a2 '&' c2) =('not' a2 'or' 'not' b2) '&'
  'not'( a2 '&' c2) by BVFUNC_1:14
    .=('not' b2 'or' 'not' a2) '&' ('not' c2 'or' 'not' a2) by BVFUNC_1:14
    .=(b2 'imp' 'not' a2) '&' ('not' c2 'or' 'not' a2) by BVFUNC_4:8
    .=(b2 'imp' 'not' a2) '&' (c2 'imp' 'not' a2) by BVFUNC_4:8
    .=(b2 'or' c2) 'imp' 'not' a2 by Th75;
  then (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2
'&' b2) '&' 'not'( a2 '&' c2) 'imp' ((b1 'or' c1) 'imp' (b2 'or' c2)) '&' ((b2
  'or' c2) 'imp' 'not' a2) '&' ((b1 'or' c1) 'imp' 'not' a2) = I_el(Y) by A6,
Th12;
  then
A7: (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2
'&' b2) '&' 'not'( a2 '&' c2) 'imp' ((b1 'or' c1) 'imp' 'not' a2) = I_el(Y) by
A1,BVFUNC_5:9;
  (a1 'or' b1 'or' c1) '&' ((b1 'or' c1) 'imp' 'not' a2) =(a1 'or' (b1
'or' c1) ) '&' ((b1 'or' c1) 'imp' 'not' a2) & (a1 'or' (b1 'or' c1)) '&' ((b1
  'or' c1) 'imp' 'not' a2) '<' (a1 'or' 'not' a2) by Th1,BVFUNC_1:8;
  then
A8: (a1 'or' b1 'or' c1) '&' ((b1 'or' c1) 'imp' 'not' a2) 'imp' (a1 'or'
  'not' a2) =I_el(Y) by BVFUNC_1:16;
  (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2
  '&' b2) '&' 'not'( a2 '&' c2) 'imp' (a1 'or' b1 'or' c1) = I_el(Y) by Lm3,
BVFUNC_1:16;
  then (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2
  '&' b2) '&' 'not'( a2 '&' c2) 'imp' (a1 'or' b1 'or' c1) '&' ((b1 'or' c1)
  'imp' 'not' a2) = I_el(Y) by A7,th18;
  then (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2
  '&' b2) '&' 'not'( a2 '&' c2) 'imp' (a1 'or' 'not' a2) = I_el(Y) by A8,
BVFUNC_5:9;
  then (b1 'imp' b2) '&' (c1 'imp' c2) '&' (a1 'or' b1 'or' c1) '&' 'not'( a2
  '&' b2) '&' 'not'( a2 '&' c2) 'imp' (a2 'imp' a1) = I_el(Y) by BVFUNC_4:8;
  hence thesis by BVFUNC_1:16;
end;
