let x, y, c be set ; :: thesis: for s being State of (MajorityICirc x,y,c)
for a, b being Element of BOOLEAN st a = s . x & b = s . y holds
(Following s) . [<*x,y*>,'&' ] = a '&' b
set xy = <*x,y*>;
set S1 = 1GateCircStr <*x,y*>,'&' ;
set A1 = 1GateCircuit x,y,'&' ;
reconsider xx = x, yy = y as Vertex of (1GateCircStr <*x,y*>,'&' ) by Th43;
reconsider v1 = [<*x,y*>,'&' ] as Element of InnerVertices (1GateCircStr <*x,y*>,'&' ) by Th47;
set S2 = 1GateCircStr <*y,c*>,'&' ;
set A2 = 1GateCircuit y,c,'&' ;
set S3 = 1GateCircStr <*c,x*>,'&' ;
set A3 = 1GateCircuit c,x,'&' ;
set S = MajorityIStr x,y,c;
set A = MajorityICirc x,y,c;
let s be State of (MajorityICirc x,y,c); :: thesis: for a, b being Element of BOOLEAN st a = s . x & b = s . y holds
(Following s) . [<*x,y*>,'&' ] = a '&' b
let a, b be Element of BOOLEAN ; :: thesis: ( a = s . x & b = s . y implies (Following s) . [<*x,y*>,'&' ] = a '&' b )
assume A1: ( a = s . x & b = s . y ) ; :: thesis: (Following s) . [<*x,y*>,'&' ] = a '&' b
A2: MajorityICirc x,y,c = (1GateCircuit x,y,'&' ) +* ((1GateCircuit y,c,'&' ) +* (1GateCircuit c,x,'&' )) by Th25;
then reconsider s1 = s | the carrier of (1GateCircStr <*x,y*>,'&' ) as State of (1GateCircuit x,y,'&' ) by Th26;
A3: MajorityIStr x,y,c = (1GateCircStr <*x,y*>,'&' ) +* ((1GateCircStr <*y,c*>,'&' ) +* (1GateCircStr <*c,x*>,'&' )) by CIRCCOMB:10;
then reconsider v = v1 as Element of InnerVertices (MajorityIStr x,y,c) by Th21;
reconsider xx = xx, yy = yy as Vertex of (MajorityIStr x,y,c) by A3, Th20;
A4: dom s1 = the carrier of (1GateCircStr <*x,y*>,'&' ) by CIRCUIT1:4;
thus (Following s) . [<*x,y*>,'&' ] = (Following s1) . v by A3, A2, CIRCCOMB:72
.= '&' . <*(s1 . xx),(s1 . yy)*> by Th48
.= '&' . <*(s . xx),(s1 . yy)*> by A4, FUNCT_1:70
.= '&' . <*(s . xx),(s . yy)*> by A4, FUNCT_1:70
.= a '&' b by A1, Def6 ; :: thesis: verum