let Y be non empty set ; :: thesis: for a, b being Element of Funcs Y,BOOLEAN holds a '<' (a 'imp' b) 'eqv' b
let a, b be Element of Funcs Y,BOOLEAN ; :: thesis: a '<' (a 'imp' b) 'eqv' b
let z be Element of Y; :: according to BVFUNC_1:def 15 :: thesis: ( not K90(Y,BOOLEAN ,a,z) = TRUE or K90(Y,BOOLEAN ,((a 'imp' b) 'eqv' b),z) = TRUE )
assume A1: a . z = TRUE ; :: thesis: K90(Y,BOOLEAN ,((a 'imp' b) 'eqv' b),z) = TRUE
then A2: 'not' (a . z) = FALSE by MARGREL1:41;
((a 'imp' b) 'eqv' b) . z = ((('not' a) 'or' b) 'eqv' b) . z by BVFUNC_4:8
.= (((('not' a) 'or' b) 'imp' b) '&' (b 'imp' (('not' a) 'or' b))) . z by BVFUNC_4:7
.= ((('not' (('not' a) 'or' b)) 'or' b) '&' (b 'imp' (('not' a) 'or' b))) . z by BVFUNC_4:8
.= ((('not' (('not' a) 'or' b)) 'or' b) '&' (('not' b) 'or' (('not' a) 'or' b))) . z by BVFUNC_4:8
.= ((('not' (('not' a) 'or' b)) 'or' b) . z) '&' ((('not' b) 'or' (('not' a) 'or' b)) . z) by MARGREL1:def 21
.= ((('not' (('not' a) 'or' b)) . z) 'or' (b . z)) '&' ((('not' b) 'or' (('not' a) 'or' b)) . z) by BVFUNC_1:def 7
.= (('not' ((('not' a) 'or' b) . z)) 'or' (b . z)) '&' ((('not' b) 'or' (('not' a) 'or' b)) . z) by MARGREL1:def 20
.= (('not' ((('not' a) . z) 'or' (b . z))) 'or' (b . z)) '&' ((('not' b) 'or' (('not' a) 'or' b)) . z) by BVFUNC_1:def 7
.= ((('not' ('not' (a . z))) '&' ('not' (b . z))) 'or' (b . z)) '&' ((('not' b) 'or' (('not' a) 'or' b)) . z) by MARGREL1:def 20
.= (((a . z) '&' ('not' (b . z))) 'or' (b . z)) '&' ((('not' b) . z) 'or' ((('not' a) 'or' b) . z)) by BVFUNC_1:def 7
.= (((a . z) '&' ('not' (b . z))) 'or' (b . z)) '&' ((('not' b) . z) 'or' ((('not' a) . z) 'or' (b . z))) by BVFUNC_1:def 7
.= (((a . z) '&' ('not' (b . z))) 'or' (b . z)) '&' ((('not' b) . z) 'or' (('not' (a . z)) 'or' (b . z))) by MARGREL1:def 20
.= ((TRUE '&' ('not' (b . z))) 'or' (b . z)) '&' (('not' (b . z)) 'or' (FALSE 'or' (b . z))) by A1, A2, MARGREL1:def 20
.= (('not' (b . z)) 'or' (b . z)) '&' (('not' (b . z)) 'or' (FALSE 'or' (b . z))) by MARGREL1:50
.= (('not' (b . z)) 'or' (b . z)) '&' (('not' (b . z)) 'or' (b . z)) by BINARITH:7
.= TRUE by XBOOLEAN:102 ;
hence K90(Y,BOOLEAN ,((a 'imp' b) 'eqv' b),z) = TRUE ; :: thesis: verum