let Y be non empty set ; :: thesis: for a1, b1, c1, a2, b2, c2 being Element of Funcs Y,BOOLEAN holds (((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ('not' (a2 '&' b2))) '&' ('not' (a2 '&' c2))) '&' ('not' (b2 '&' c2))) 'imp' (((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' ((a1 'or' b1) 'or' c1)) '&' ('not' (c2 '&' a2))) '&' ('not' (c2 '&' b2))) = I_el Y
let a1, b1, c1, a2, b2, c2 be Element of Funcs Y,BOOLEAN ; :: thesis: (((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ('not' (a2 '&' b2))) '&' ('not' (a2 '&' c2))) '&' ('not' (b2 '&' c2))) 'imp' (((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' ((a1 'or' b1) 'or' c1)) '&' ('not' (c2 '&' a2))) '&' ('not' (c2 '&' b2))) = I_el Y
A1: (((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ('not' (a2 '&' b2))) '&' ('not' (a2 '&' c2))) '&' ('not' (b2 '&' c2))) 'imp' (a1 'imp' a2) = I_el Y by Lm7;
A2: (((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ('not' (a2 '&' b2))) '&' ('not' (a2 '&' c2))) '&' ('not' (b2 '&' c2))) 'imp' (b1 'imp' b2) = I_el Y by Lm7;
A3: (((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ('not' (a2 '&' b2))) '&' ('not' (a2 '&' c2))) '&' ('not' (b2 '&' c2))) 'imp' ((a1 'or' b1) 'or' c1) = I_el Y by Lm7;
A4: (((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ('not' (a2 '&' b2))) '&' ('not' (a2 '&' c2))) '&' ('not' (b2 '&' c2))) 'imp' ('not' (a2 '&' c2)) = I_el Y by Lm7;
A5: (((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ('not' (a2 '&' b2))) '&' ('not' (a2 '&' c2))) '&' ('not' (b2 '&' c2))) 'imp' ('not' (b2 '&' c2)) = I_el Y by Lm7;
(((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ('not' (a2 '&' b2))) '&' ('not' (a2 '&' c2))) '&' ('not' (b2 '&' c2))) 'imp' ((a1 'imp' a2) '&' (b1 'imp' b2)) = I_el Y by A1, A2, BVFUNC_6:18;
then (((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ('not' (a2 '&' b2))) '&' ('not' (a2 '&' c2))) '&' ('not' (b2 '&' c2))) 'imp' (((a1 'imp' a2) '&' (b1 'imp' b2)) '&' ((a1 'or' b1) 'or' c1)) = I_el Y by A3, BVFUNC_6:18;
then (((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ('not' (a2 '&' b2))) '&' ('not' (a2 '&' c2))) '&' ('not' (b2 '&' c2))) 'imp' ((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' ((a1 'or' b1) 'or' c1)) '&' ('not' (a2 '&' c2))) = I_el Y by A4, BVFUNC_6:18;
hence (((((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' (c1 'imp' c2)) '&' ((a1 'or' b1) 'or' c1)) '&' ('not' (a2 '&' b2))) '&' ('not' (a2 '&' c2))) '&' ('not' (b2 '&' c2))) 'imp' (((((a1 'imp' a2) '&' (b1 'imp' b2)) '&' ((a1 'or' b1) 'or' c1)) '&' ('not' (c2 '&' a2))) '&' ('not' (c2 '&' b2))) = I_el Y by A5, BVFUNC_6:18; :: thesis: verum