let x, y, c be object ; :: thesis: ( c <> [<*x,y*>,'or'] implies for s being State of (2GatesCircuit (x,y,c,'or'))
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,'or')) = (a1 'or' a2) 'or' a3 )
set f = 'or' ;
assume A1: c <> [<*x,y*>,'or'] ; :: thesis: for s being State of (2GatesCircuit (x,y,c,'or'))
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,'or')) = (a1 'or' a2) 'or' a3
let s be State of (2GatesCircuit (x,y,c,'or')); :: 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,'or')) = (a1 'or' a2) 'or' 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,'or')) = (a1 'or' a2) 'or' a3 )
assume ( a1 = s . x & a2 = s . y & a3 = s . c ) ; :: thesis: (Following (s,2)) . (2GatesCircOutput (x,y,c,'or')) = (a1 'or' a2) 'or' a3
hence (Following (s,2)) . (2GatesCircOutput (x,y,c,'or')) = 'or' . <*('or' . <*a1,a2*>),a3*> by A1, Th62
.= 'or' . <*(a1 'or' a2),a3*> by Def4
.= (a1 'or' a2) 'or' a3 by Def4 ;
:: thesis: verum