let x, y, c be set ; :: thesis: for s being State of (MajorityCirc x,y,c)
for a1, a2, a3 being Element of BOOLEAN st a1 = s . [<*x,y*>,'&' ] & a2 = s . [<*y,c*>,'&' ] & a3 = s . [<*c,x*>,'&' ] holds
(Following s) . (MajorityOutput x,y,c) = (a1 'or' a2) 'or' a3
let s be State of (MajorityCirc x,y,c); :: thesis: for a1, a2, a3 being Element of BOOLEAN st a1 = s . [<*x,y*>,'&' ] & a2 = s . [<*y,c*>,'&' ] & a3 = s . [<*c,x*>,'&' ] holds
(Following s) . (MajorityOutput x,y,c) = (a1 'or' a2) 'or' a3
let a1, a2, a3 be Element of BOOLEAN ; :: thesis: ( a1 = s . [<*x,y*>,'&' ] & a2 = s . [<*y,c*>,'&' ] & a3 = s . [<*c,x*>,'&' ] implies (Following s) . (MajorityOutput x,y,c) = (a1 'or' a2) 'or' a3 )
assume that
A1: a1 = s . [<*x,y*>,'&' ] and
A2: a2 = s . [<*y,c*>,'&' ] and
A3: a3 = s . [<*c,x*>,'&' ] ; :: thesis: (Following s) . (MajorityOutput x,y,c) = (a1 'or' a2) 'or' a3
set xy = [<*x,y*>,'&' ];
set yc = [<*y,c*>,'&' ];
set cx = [<*c,x*>,'&' ];
set S = MajorityStr x,y,c;
A4: InnerVertices (MajorityStr x,y,c) = the carrier' of (MajorityStr x,y,c) by FACIRC_1:37;
A5: dom s = the carrier of (MajorityStr x,y,c) by CIRCUIT1:4;
reconsider xy = [<*x,y*>,'&' ], yc = [<*y,c*>,'&' ], cx = [<*c,x*>,'&' ] as Element of InnerVertices (MajorityStr x,y,c) by FACIRC_1:73;
thus (Following s) . (MajorityOutput x,y,c) = or3 . (s * <*xy,yc,cx*>) by A4, FACIRC_1:35
.= or3 . <*a1,a2,a3*> by A1, A2, A3, A5, FINSEQ_2:146
.= (a1 'or' a2) 'or' a3 by FACIRC_1:def 7 ; :: thesis: verum