let c, x, y be set ; :: thesis: ( c <> [<*x,y*>,'&' ] implies for s being State of (2GatesCircuit x,y,c,'&' )
for a1, a2, a3 being Element of BOOLEAN st a1 = s . x & a2 = s . y & a3 = s . c holds
(Following s,2) . (2GatesCircOutput x,y,c,'&' ) = (a1 '&' a2) '&' a3 )
set f = '&' ;
assume A1: c <> [<*x,y*>,'&' ] ; :: thesis: for s being State of (2GatesCircuit x,y,c,'&' )
for a1, a2, a3 being Element of BOOLEAN st a1 = s . x & a2 = s . y & a3 = s . c holds
(Following s,2) . (2GatesCircOutput x,y,c,'&' ) = (a1 '&' a2) '&' a3
let s be State of (2GatesCircuit x,y,c,'&' ); :: thesis: for a1, a2, a3 being Element of BOOLEAN st a1 = s . x & a2 = s . y & a3 = s . c holds
(Following s,2) . (2GatesCircOutput x,y,c,'&' ) = (a1 '&' a2) '&' a3
let a1, a2, a3 be Element of BOOLEAN ; :: thesis: ( a1 = s . x & a2 = s . y & a3 = s . c implies (Following s,2) . (2GatesCircOutput x,y,c,'&' ) = (a1 '&' a2) '&' a3 )
assume ( a1 = s . x & a2 = s . y & a3 = s . c ) ; :: thesis: (Following s,2) . (2GatesCircOutput x,y,c,'&' ) = (a1 '&' a2) '&' a3
hence (Following s,2) . (2GatesCircOutput x,y,c,'&' ) = '&' . <*('&' . <*a1,a2*>),a3*> by A1, Th62
.= '&' . <*(a1 '&' a2),a3*> by Def6
.= (a1 '&' a2) '&' a3 by Def6 ;
:: thesis: verum