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 & 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: 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;
A6: x in the carrier of (MajorityStr (x,y,c)) by FACIRC_1:72;
A7: y in the carrier of (MajorityStr (x,y,c)) by FACIRC_1:72;
A8: c in the carrier of (MajorityStr (x,y,c)) by FACIRC_1:72;
[<*x,y*>,'&'] in InnerVertices (MajorityStr (x,y,c)) by FACIRC_1:73;
hence (Following s) . [<*x,y*>,'&'] = '&' . (s * <*x,y*>) by A4, FACIRC_1:35
.= '&' . <*a1,a2*> by A1, A2, A5, A6, A7, FINSEQ_2:125
.= a1 '&' a2 by FACIRC_1:def 6 ;
:: thesis: ( (Following s) . [<*y,c*>,'&'] = a2 '&' a3 & (Following s) . [<*c,x*>,'&'] = a3 '&' a1 )
[<*y,c*>,'&'] in InnerVertices (MajorityStr (x,y,c)) by FACIRC_1:73;
hence (Following s) . [<*y,c*>,'&'] = '&' . (s * <*y,c*>) by A4, FACIRC_1:35
.= '&' . <*a2,a3*> by A2, A3, A5, A7, A8, FINSEQ_2:125
.= a2 '&' a3 by FACIRC_1:def 6 ;
:: thesis: (Following s) . [<*c,x*>,'&'] = a3 '&' a1
[<*c,x*>,'&'] in InnerVertices (MajorityStr (x,y,c)) by FACIRC_1:73;
hence (Following s) . [<*c,x*>,'&'] = '&' . (s * <*c,x*>) by A4, FACIRC_1:35
.= '&' . <*a3,a1*> by A1, A3, A5, A6, A8, FINSEQ_2:125
.= a3 '&' a1 by FACIRC_1:def 6 ;
:: thesis: verum