let x, y, c be object ; :: thesis: for s being State of (MajorityICirc (x,y,c))
for a, b being Element of BOOLEAN st a = s . y & b = s . c holds
(Following s) . [<*y,c*>,'&'] = a '&' b
set yc = <*y,c*>;
set S2 = 1GateCircStr (<*y,c*>,'&');
set A2 = 1GateCircuit (y,c,'&');
reconsider yy = y, cc = c as Vertex of (1GateCircStr (<*y,c*>,'&')) by Th43;
reconsider v2 = [<*y,c*>,'&'] as Element of InnerVertices (1GateCircStr (<*y,c*>,'&')) by Th47;
set S1 = 1GateCircStr (<*x,y*>,'&');
set A1 = 1GateCircuit (x,y,'&');
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 . y & b = s . c holds
(Following s) . [<*y,c*>,'&'] = a '&' b
let a, b be Element of BOOLEAN ; :: thesis: ( a = s . y & b = s . c implies (Following s) . [<*y,c*>,'&'] = a '&' b )
assume A1: ( a = s . y & b = s . c ) ; :: thesis: (Following s) . [<*y,c*>,'&'] = a '&' b
A2: (1GateCircStr (<*x,y*>,'&')) +* (1GateCircStr (<*y,c*>,'&')) = (1GateCircStr (<*y,c*>,'&')) +* (1GateCircStr (<*x,y*>,'&')) by CIRCCOMB:5, CIRCCOMB:47;
then A3: MajorityIStr (x,y,c) = (1GateCircStr (<*y,c*>,'&')) +* ((1GateCircStr (<*x,y*>,'&')) +* (1GateCircStr (<*c,x*>,'&'))) by CIRCCOMB:6;
then reconsider v = v2 as Element of InnerVertices (MajorityIStr (x,y,c)) by Th21;
(1GateCircuit (x,y,'&')) +* (1GateCircuit (y,c,'&')) = (1GateCircuit (y,c,'&')) +* (1GateCircuit (x,y,'&')) by CIRCCOMB:22, CIRCCOMB:60;
then A4: MajorityICirc (x,y,c) = (1GateCircuit (y,c,'&')) +* ((1GateCircuit (x,y,'&')) +* (1GateCircuit (c,x,'&'))) by A2, Th25;
then reconsider s2 = s | the carrier of (1GateCircStr (<*y,c*>,'&')) as State of (1GateCircuit (y,c,'&')) by Th26;
reconsider yy = yy, cc = cc as Vertex of (MajorityIStr (x,y,c)) by A3, Th20;
A5: dom s2 = the carrier of (1GateCircStr (<*y,c*>,'&')) by CIRCUIT1:3;
thus (Following s) . [<*y,c*>,'&'] = (Following s2) . v by A3, A4, CIRCCOMB:64
.= '&' . <*(s2 . yy),(s2 . cc)*> by Th48
.= '&' . <*(s . yy),(s2 . cc)*> by A5, FUNCT_1:47
.= '&' . <*(s . yy),(s . cc)*> by A5, FUNCT_1:47
.= a '&' b by A1, Def5 ; :: thesis: verum