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 CIRCUIT1:3;
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_2:126
.= (a1 'or' a2) 'or' a3 by FACIRC_1:def 7 ; :: thesis: verum