let x, y, c be non pair object ; :: 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

set xy = [<*x,y*>,'&'];

set yc = [<*y,c*>,'&'];

set cx = [<*c,x*>,'&'];

set S = MajorityStr (x,y,c);

reconsider xy = [<*x,y*>,'&'], yc = [<*y,c*>,'&'], cx = [<*c,x*>,'&'] as Element of InnerVertices (MajorityStr (x,y,c)) by Th73;

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 A1: ( a1 = s . [<*x,y*>,'&'] & a2 = s . [<*y,c*>,'&'] & a3 = s . [<*c,x*>,'&'] ) ; :: thesis: (Following s) . (MajorityOutput (x,y,c)) = (a1 'or' a2) 'or' a3

A2: dom s = the carrier of (MajorityStr (x,y,c)) by CIRCUIT1:3;

InnerVertices (MajorityStr (x,y,c)) = the carrier' of (MajorityStr (x,y,c)) by Th37;

hence (Following s) . (MajorityOutput (x,y,c)) = or3 . (s * <*xy,yc,cx*>) by Th35

.= or3 . <*a1,a2,a3*> by A1, A2, FINSEQ_2:126

.= (a1 'or' a2) 'or' a3 by Def6 ;

:: thesis: verum

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

set xy = [<*x,y*>,'&'];

set yc = [<*y,c*>,'&'];

set cx = [<*c,x*>,'&'];

set S = MajorityStr (x,y,c);

reconsider xy = [<*x,y*>,'&'], yc = [<*y,c*>,'&'], cx = [<*c,x*>,'&'] as Element of InnerVertices (MajorityStr (x,y,c)) by Th73;

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 A1: ( a1 = s . [<*x,y*>,'&'] & a2 = s . [<*y,c*>,'&'] & a3 = s . [<*c,x*>,'&'] ) ; :: thesis: (Following s) . (MajorityOutput (x,y,c)) = (a1 'or' a2) 'or' a3

A2: dom s = the carrier of (MajorityStr (x,y,c)) by CIRCUIT1:3;

InnerVertices (MajorityStr (x,y,c)) = the carrier' of (MajorityStr (x,y,c)) by Th37;

hence (Following s) . (MajorityOutput (x,y,c)) = or3 . (s * <*xy,yc,cx*>) by Th35

.= or3 . <*a1,a2,a3*> by A1, A2, FINSEQ_2:126

.= (a1 'or' a2) 'or' a3 by Def6 ;

:: thesis: verum