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

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

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

assume that

A1: a1 = s . x and

A2: a2 = s . y and

A3: a3 = s . c ; :: thesis: ( (Following s) . [<*x,y*>,'&'] = a1 '&' a2 & (Following s) . [<*y,c*>,'&'] = a2 '&' a3 & (Following s) . [<*c,x*>,'&'] = a3 '&' a1 )

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

A4: dom s = the carrier of (MajorityStr (x,y,c)) by CIRCUIT1:3;

A5: y in the carrier of (MajorityStr (x,y,c)) by Th72;

A6: x in the carrier of (MajorityStr (x,y,c)) by Th72;

A7: InnerVertices (MajorityStr (x,y,c)) = the carrier' of (MajorityStr (x,y,c)) by Th37;

[<*x,y*>,'&'] in InnerVertices (MajorityStr (x,y,c)) by Th73;

hence (Following s) . [<*x,y*>,'&'] = '&' . (s * <*x,y*>) by A7, Th35

.= '&' . <*a1,a2*> by A1, A2, A4, A6, A5, FINSEQ_2:125

.= a1 '&' a2 by Def5 ;

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

A8: c in the carrier of (MajorityStr (x,y,c)) by Th72;

[<*y,c*>,'&'] in InnerVertices (MajorityStr (x,y,c)) by Th73;

hence (Following s) . [<*y,c*>,'&'] = '&' . (s * <*y,c*>) by A7, Th35

.= '&' . <*a2,a3*> by A2, A3, A4, A5, A8, FINSEQ_2:125

.= a2 '&' a3 by Def5 ;

:: thesis: (Following s) . [<*c,x*>,'&'] = a3 '&' a1

[<*c,x*>,'&'] in InnerVertices (MajorityStr (x,y,c)) by Th73;

hence (Following s) . [<*c,x*>,'&'] = '&' . (s * <*c,x*>) by A7, Th35

.= '&' . <*a3,a1*> by A1, A3, A4, A6, A8, FINSEQ_2:125

.= a3 '&' a1 by Def5 ;

:: thesis: verum

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

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

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

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

assume that

A1: a1 = s . x and

A2: a2 = s . y and

A3: a3 = s . c ; :: thesis: ( (Following s) . [<*x,y*>,'&'] = a1 '&' a2 & (Following s) . [<*y,c*>,'&'] = a2 '&' a3 & (Following s) . [<*c,x*>,'&'] = a3 '&' a1 )

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

A4: dom s = the carrier of (MajorityStr (x,y,c)) by CIRCUIT1:3;

A5: y in the carrier of (MajorityStr (x,y,c)) by Th72;

A6: x in the carrier of (MajorityStr (x,y,c)) by Th72;

A7: InnerVertices (MajorityStr (x,y,c)) = the carrier' of (MajorityStr (x,y,c)) by Th37;

[<*x,y*>,'&'] in InnerVertices (MajorityStr (x,y,c)) by Th73;

hence (Following s) . [<*x,y*>,'&'] = '&' . (s * <*x,y*>) by A7, Th35

.= '&' . <*a1,a2*> by A1, A2, A4, A6, A5, FINSEQ_2:125

.= a1 '&' a2 by Def5 ;

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

A8: c in the carrier of (MajorityStr (x,y,c)) by Th72;

[<*y,c*>,'&'] in InnerVertices (MajorityStr (x,y,c)) by Th73;

hence (Following s) . [<*y,c*>,'&'] = '&' . (s * <*y,c*>) by A7, Th35

.= '&' . <*a2,a3*> by A2, A3, A4, A5, A8, FINSEQ_2:125

.= a2 '&' a3 by Def5 ;

:: thesis: (Following s) . [<*c,x*>,'&'] = a3 '&' a1

[<*c,x*>,'&'] in InnerVertices (MajorityStr (x,y,c)) by Th73;

hence (Following s) . [<*c,x*>,'&'] = '&' . (s * <*c,x*>) by A7, Th35

.= '&' . <*a3,a1*> by A1, A3, A4, A6, A8, FINSEQ_2:125

.= a3 '&' a1 by Def5 ;

:: thesis: verum