let Y be non empty set ; :: thesis: for a, b being Function of Y,BOOLEAN holds a '<' ((a 'or' b) 'eqv' (b 'or' a)) 'eqv' a
let a, b be Function of Y,BOOLEAN; :: thesis: a '<' ((a 'or' b) 'eqv' (b 'or' a)) 'eqv' a
let z be Element of Y; :: according to BVFUNC_1:def 12 :: thesis: ( not a . z = TRUE or (((a 'or' b) 'eqv' (b 'or' a)) 'eqv' a) . z = TRUE )
assume A1: a . z = TRUE ; :: thesis: (((a 'or' b) 'eqv' (b 'or' a)) 'eqv' a) . z = TRUE
A2: ((a 'or' b) 'eqv' (b 'or' a)) . z = (((a 'or' b) 'imp' (a 'or' b)) '&' ((a 'or' b) 'imp' (a 'or' b))) . z by BVFUNC_4:7
.= (('not' (a 'or' b)) 'or' (a 'or' b)) . z by BVFUNC_4:8
.= (I_el Y) . z by BVFUNC_4:6
.= TRUE by BVFUNC_1:def 11 ;
(((a 'or' b) 'eqv' (b 'or' a)) 'eqv' a) . z = ((((a 'or' b) 'eqv' (a 'or' b)) 'imp' a) '&' (a 'imp' ((a 'or' b) 'eqv' (a 'or' b)))) . z by BVFUNC_4:7
.= ((((a 'or' b) 'eqv' (a 'or' b)) 'imp' a) . z) '&' ((a 'imp' ((a 'or' b) 'eqv' (a 'or' b))) . z) by MARGREL1:def 20
.= ((('not' ((a 'or' b) 'eqv' (a 'or' b))) 'or' a) . z) '&' ((a 'imp' ((a 'or' b) 'eqv' (a 'or' b))) . z) by BVFUNC_4:8
.= ((('not' ((a 'or' b) 'eqv' (a 'or' b))) 'or' a) . z) '&' ((('not' a) 'or' ((a 'or' b) 'eqv' (a 'or' b))) . z) by BVFUNC_4:8
.= ((('not' ((a 'or' b) 'eqv' (a 'or' b))) . z) 'or' (a . z)) '&' ((('not' a) 'or' ((a 'or' b) 'eqv' (a 'or' b))) . z) by BVFUNC_1:def 4
.= ((('not' ((a 'or' b) 'eqv' (a 'or' b))) . z) 'or' (a . z)) '&' ((('not' a) . z) 'or' (((a 'or' b) 'eqv' (a 'or' b)) . z)) by BVFUNC_1:def 4
.= (('not' (((a 'or' b) 'eqv' (a 'or' b)) . z)) 'or' (a . z)) '&' ((('not' a) . z) 'or' (((a 'or' b) 'eqv' (a 'or' b)) . z)) by MARGREL1:def 19
.= (FALSE 'or' (a . z)) '&' ((('not' a) . z) 'or' TRUE) by A2
.= (a . z) '&' ((('not' a) . z) 'or' TRUE)
.= (a . z) '&' TRUE
.= TRUE by A1 ;
hence (((a 'or' b) 'eqv' (b 'or' a)) 'eqv' a) . z = TRUE ; :: thesis: verum