let x, y, c be object ; :: 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 Def5
.= (a1 '&' a2) '&' a3 by Def5 ;
:: thesis: verum