let Y be non empty set ; :: thesis: for a, b being Element of Funcs Y,BOOLEAN holds 'not' a '<' (a 'imp' b) 'eqv' ('not' a)
let a, b be Element of Funcs Y,BOOLEAN ; :: thesis: 'not' a '<' (a 'imp' b) 'eqv' ('not' a)
let z be Element of Y; :: according to BVFUNC_1:def 15 :: thesis: ( not K90(Y,BOOLEAN ,('not' a),z) = TRUE or K90(Y,BOOLEAN ,((a 'imp' b) 'eqv' ('not' a)),z) = TRUE )
assume A1: ('not' a) . z = TRUE ; :: thesis: K90(Y,BOOLEAN ,((a 'imp' b) 'eqv' ('not' a)),z) = TRUE
then 'not' (a . z) = TRUE by MARGREL1:def 20;
then A2: a . z = FALSE by MARGREL1:41;
((a 'imp' b) 'eqv' ('not' a)) . z = ((('not' a) 'or' b) 'eqv' ('not' a)) . z by BVFUNC_4:8
.= (((('not' a) 'or' b) 'imp' ('not' a)) '&' (('not' a) 'imp' (('not' a) 'or' b))) . z by BVFUNC_4:7
.= ((('not' (('not' a) 'or' b)) 'or' ('not' a)) '&' (('not' a) 'imp' (('not' a) 'or' b))) . z by BVFUNC_4:8
.= ((('not' (('not' a) 'or' b)) 'or' ('not' a)) '&' (('not' ('not' a)) 'or' (('not' a) 'or' b))) . z by BVFUNC_4:8
.= ((('not' (('not' a) 'or' b)) 'or' ('not' a)) . z) '&' ((('not' ('not' a)) 'or' (('not' a) 'or' b)) . z) by MARGREL1:def 21
.= ((('not' (('not' a) 'or' b)) . z) 'or' (('not' a) . z)) '&' ((('not' ('not' a)) 'or' (('not' a) 'or' b)) . z) by BVFUNC_1:def 7
.= (('not' ((('not' a) 'or' b) . z)) 'or' (('not' a) . z)) '&' ((('not' ('not' a)) 'or' (('not' a) 'or' b)) . z) by MARGREL1:def 20
.= (('not' ((('not' a) . z) 'or' (b . z))) 'or' (('not' a) . z)) '&' ((('not' ('not' a)) 'or' (('not' a) 'or' b)) . z) by BVFUNC_1:def 7
.= ((('not' ('not' (a . z))) '&' ('not' (b . z))) 'or' (('not' a) . z)) '&' ((('not' ('not' a)) 'or' (('not' a) 'or' b)) . z) by MARGREL1:def 20
.= (((a . z) '&' ('not' (b . z))) 'or' (('not' a) . z)) '&' ((('not' ('not' a)) . z) 'or' ((('not' a) 'or' b) . z)) by BVFUNC_1:def 7
.= (((a . z) '&' ('not' (b . z))) 'or' (('not' a) . z)) '&' ((a . z) 'or' ((('not' a) . z) 'or' (b . z))) by BVFUNC_1:def 7
.= TRUE '&' (FALSE 'or' (TRUE 'or' (b . z))) by A1, A2, BINARITH:19
.= FALSE 'or' (TRUE 'or' (b . z)) by MARGREL1:50
.= TRUE 'or' (b . z) by BINARITH:7
.= TRUE by BINARITH:19 ;
hence K90(Y,BOOLEAN ,((a 'imp' b) 'eqv' ('not' a)),z) = TRUE ; :: thesis: verum