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 & a2 = s . y & a3 = s . c holds

( (Following (s,2)) . (MajorityOutput (x,y,c)) = ((a1 '&' a2) 'or' (a2 '&' a3)) 'or' (a3 '&' a1) & (Following (s,2)) . [<*x,y*>,'&'] = a1 '&' a2 & (Following (s,2)) . [<*y,c*>,'&'] = a2 '&' a3 & (Following (s,2)) . [<*c,x*>,'&'] = a3 '&' a1 )

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

reconsider x9 = x, y9 = y, c9 = c as Vertex of (MajorityStr (x,y,c)) by Th72;

let s be State of (MajorityCirc (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)) . (MajorityOutput (x,y,c)) = ((a1 '&' a2) 'or' (a2 '&' a3)) 'or' (a3 '&' a1) & (Following (s,2)) . [<*x,y*>,'&'] = a1 '&' a2 & (Following (s,2)) . [<*y,c*>,'&'] = a2 '&' a3 & (Following (s,2)) . [<*c,x*>,'&'] = a3 '&' a1 )

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

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

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

A1: Following (s,2) = Following (Following s) by Th15;

let a1, a2, a3 be Element of BOOLEAN ; :: thesis: ( a1 = s . x & a2 = s . y & a3 = s . c implies ( (Following (s,2)) . (MajorityOutput (x,y,c)) = ((a1 '&' a2) 'or' (a2 '&' a3)) 'or' (a3 '&' a1) & (Following (s,2)) . [<*x,y*>,'&'] = a1 '&' a2 & (Following (s,2)) . [<*y,c*>,'&'] = a2 '&' a3 & (Following (s,2)) . [<*c,x*>,'&'] = a3 '&' a1 ) )

assume A2: ( a1 = s . x & a2 = s . y & a3 = s . c ) ; :: thesis: ( (Following (s,2)) . (MajorityOutput (x,y,c)) = ((a1 '&' a2) 'or' (a2 '&' a3)) 'or' (a3 '&' a1) & (Following (s,2)) . [<*x,y*>,'&'] = a1 '&' a2 & (Following (s,2)) . [<*y,c*>,'&'] = a2 '&' a3 & (Following (s,2)) . [<*c,x*>,'&'] = a3 '&' a1 )

A3: (Following s) . [<*c,x*>,'&'] = a3 '&' a1 by A2, Lm2;

( (Following s) . [<*x,y*>,'&'] = a1 '&' a2 & (Following s) . [<*y,c*>,'&'] = a2 '&' a3 ) by A2, Lm2;

hence (Following (s,2)) . (MajorityOutput (x,y,c)) = ((a1 '&' a2) 'or' (a2 '&' a3)) 'or' (a3 '&' a1) by A1, A3, Th79; :: thesis: ( (Following (s,2)) . [<*x,y*>,'&'] = a1 '&' a2 & (Following (s,2)) . [<*y,c*>,'&'] = a2 '&' a3 & (Following (s,2)) . [<*c,x*>,'&'] = a3 '&' a1 )

y in InputVertices (MajorityStr (x,y,c)) by Th74;

then A4: (Following s) . y9 = s . y by CIRCUIT2:def 5;

c in InputVertices (MajorityStr (x,y,c)) by Th74;

then A5: (Following s) . c9 = s . c by CIRCUIT2:def 5;

x in InputVertices (MajorityStr (x,y,c)) by Th74;

then (Following s) . x9 = s . x by CIRCUIT2:def 5;

hence ( (Following (s,2)) . [<*x,y*>,'&'] = a1 '&' a2 & (Following (s,2)) . [<*y,c*>,'&'] = a2 '&' a3 & (Following (s,2)) . [<*c,x*>,'&'] = a3 '&' a1 ) by A2, A4, A5, A1, Lm2; :: thesis: verum

for a1, a2, a3 being Element of BOOLEAN st a1 = s . x & a2 = s . y & a3 = s . c holds

( (Following (s,2)) . (MajorityOutput (x,y,c)) = ((a1 '&' a2) 'or' (a2 '&' a3)) 'or' (a3 '&' a1) & (Following (s,2)) . [<*x,y*>,'&'] = a1 '&' a2 & (Following (s,2)) . [<*y,c*>,'&'] = a2 '&' a3 & (Following (s,2)) . [<*c,x*>,'&'] = a3 '&' a1 )

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

reconsider x9 = x, y9 = y, c9 = c as Vertex of (MajorityStr (x,y,c)) by Th72;

let s be State of (MajorityCirc (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)) . (MajorityOutput (x,y,c)) = ((a1 '&' a2) 'or' (a2 '&' a3)) 'or' (a3 '&' a1) & (Following (s,2)) . [<*x,y*>,'&'] = a1 '&' a2 & (Following (s,2)) . [<*y,c*>,'&'] = a2 '&' a3 & (Following (s,2)) . [<*c,x*>,'&'] = a3 '&' a1 )

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

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

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

A1: Following (s,2) = Following (Following s) by Th15;

let a1, a2, a3 be Element of BOOLEAN ; :: thesis: ( a1 = s . x & a2 = s . y & a3 = s . c implies ( (Following (s,2)) . (MajorityOutput (x,y,c)) = ((a1 '&' a2) 'or' (a2 '&' a3)) 'or' (a3 '&' a1) & (Following (s,2)) . [<*x,y*>,'&'] = a1 '&' a2 & (Following (s,2)) . [<*y,c*>,'&'] = a2 '&' a3 & (Following (s,2)) . [<*c,x*>,'&'] = a3 '&' a1 ) )

assume A2: ( a1 = s . x & a2 = s . y & a3 = s . c ) ; :: thesis: ( (Following (s,2)) . (MajorityOutput (x,y,c)) = ((a1 '&' a2) 'or' (a2 '&' a3)) 'or' (a3 '&' a1) & (Following (s,2)) . [<*x,y*>,'&'] = a1 '&' a2 & (Following (s,2)) . [<*y,c*>,'&'] = a2 '&' a3 & (Following (s,2)) . [<*c,x*>,'&'] = a3 '&' a1 )

A3: (Following s) . [<*c,x*>,'&'] = a3 '&' a1 by A2, Lm2;

( (Following s) . [<*x,y*>,'&'] = a1 '&' a2 & (Following s) . [<*y,c*>,'&'] = a2 '&' a3 ) by A2, Lm2;

hence (Following (s,2)) . (MajorityOutput (x,y,c)) = ((a1 '&' a2) 'or' (a2 '&' a3)) 'or' (a3 '&' a1) by A1, A3, Th79; :: thesis: ( (Following (s,2)) . [<*x,y*>,'&'] = a1 '&' a2 & (Following (s,2)) . [<*y,c*>,'&'] = a2 '&' a3 & (Following (s,2)) . [<*c,x*>,'&'] = a3 '&' a1 )

y in InputVertices (MajorityStr (x,y,c)) by Th74;

then A4: (Following s) . y9 = s . y by CIRCUIT2:def 5;

c in InputVertices (MajorityStr (x,y,c)) by Th74;

then A5: (Following s) . c9 = s . c by CIRCUIT2:def 5;

x in InputVertices (MajorityStr (x,y,c)) by Th74;

then (Following s) . x9 = s . x by CIRCUIT2:def 5;

hence ( (Following (s,2)) . [<*x,y*>,'&'] = a1 '&' a2 & (Following (s,2)) . [<*y,c*>,'&'] = a2 '&' a3 & (Following (s,2)) . [<*c,x*>,'&'] = a3 '&' a1 ) by A2, A4, A5, A1, Lm2; :: thesis: verum