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