let Y be non empty set ; :: thesis: for a, b being Element of Funcs Y,BOOLEAN holds 'not' a '<' (b 'imp' a) 'eqv' ('not' b)
let a, b be Element of Funcs Y,BOOLEAN ; :: thesis: 'not' a '<' (b 'imp' a) 'eqv' ('not' b)
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 ,((b 'imp' a) 'eqv' ('not' b)),z) = TRUE )
assume ('not' a) . z = TRUE ; :: thesis: K90(Y,BOOLEAN ,((b 'imp' a) 'eqv' ('not' b)),z) = TRUE
then A1: 'not' (a . z) = TRUE by MARGREL1:def 20;
((b 'imp' a) 'eqv' ('not' b)) . z = ((('not' b) 'or' a) 'eqv' ('not' b)) . z by BVFUNC_4:8
.= (((('not' b) 'or' a) 'imp' ('not' b)) '&' (('not' b) 'imp' (('not' b) 'or' a))) . z by BVFUNC_4:7
.= ((('not' (('not' b) 'or' a)) 'or' ('not' b)) '&' (('not' b) 'imp' (('not' b) 'or' a))) . z by BVFUNC_4:8
.= ((('not' (('not' b) 'or' a)) 'or' ('not' b)) '&' (('not' ('not' b)) 'or' (('not' b) 'or' a))) . z by BVFUNC_4:8
.= ((('not' (('not' b) 'or' a)) 'or' ('not' b)) . z) '&' ((('not' ('not' b)) 'or' (('not' b) 'or' a)) . z) by MARGREL1:def 21
.= ((('not' (('not' b) 'or' a)) . z) 'or' (('not' b) . z)) '&' ((('not' ('not' b)) 'or' (('not' b) 'or' a)) . z) by BVFUNC_1:def 7
.= (('not' ((('not' b) 'or' a) . z)) 'or' (('not' b) . z)) '&' ((('not' ('not' b)) 'or' (('not' b) 'or' a)) . z) by MARGREL1:def 20
.= (('not' ((('not' b) . z) 'or' (a . z))) 'or' (('not' b) . z)) '&' ((('not' ('not' b)) 'or' (('not' b) 'or' a)) . z) by BVFUNC_1:def 7
.= ((('not' ('not' (b . z))) '&' ('not' (a . z))) 'or' (('not' b) . z)) '&' ((('not' ('not' b)) 'or' (('not' b) 'or' a)) . z) by MARGREL1:def 20
.= (((b . z) '&' ('not' (a . z))) 'or' (('not' b) . z)) '&' ((('not' ('not' b)) . z) 'or' ((('not' b) 'or' a) . z)) by BVFUNC_1:def 7
.= (((b . z) '&' ('not' (a . z))) 'or' (('not' b) . z)) '&' ((('not' ('not' b)) . z) 'or' ((('not' b) . z) 'or' (a . z))) by BVFUNC_1:def 7
.= (((b . z) '&' ('not' (a . z))) 'or' (('not' b) . z)) '&' ((b . z) 'or' (('not' (b . z)) 'or' (a . z))) by MARGREL1:def 20
.= ((TRUE '&' (b . z)) 'or' (('not' b) . z)) '&' ((b . z) 'or' (('not' (b . z)) 'or' FALSE )) by A1, MARGREL1:41
.= ((b . z) 'or' (('not' b) . z)) '&' ((b . z) 'or' (('not' (b . z)) 'or' FALSE )) by MARGREL1:50
.= ((b . z) 'or' ('not' (b . z))) '&' ((b . z) 'or' (('not' (b . z)) 'or' FALSE )) by MARGREL1:def 20
.= TRUE '&' ((b . z) 'or' (('not' (b . z)) 'or' FALSE )) by XBOOLEAN:102
.= (b . z) 'or' (('not' (b . z)) 'or' FALSE ) by MARGREL1:50
.= ((b . z) 'or' ('not' (b . z))) 'or' FALSE by BINARITH:20
.= TRUE 'or' FALSE by XBOOLEAN:102
.= TRUE by BINARITH:19 ;
hence K90(Y,BOOLEAN ,((b 'imp' a) 'eqv' ('not' b)),z) = TRUE ; :: thesis: verum