let x, y, c be non pair 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,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 )
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 )
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 A1: ( 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 )
set xy = [<*x,y*>,'&' ];
set yc = [<*y,c*>,'&' ];
set cx = [<*c,x*>,'&' ];
set S = MajorityStr x,y,c;
reconsider x' = x, y' = y, c' = c as Vertex of (MajorityStr x,y,c) by Th72;
( x in InputVertices (MajorityStr x,y,c) & y in InputVertices (MajorityStr x,y,c) & c in InputVertices (MajorityStr x,y,c) ) by Th74;
then A2: ( (Following s) . x' = s . x & (Following s) . y' = s . y & (Following s) . c' = s . c ) by CIRCUIT2:def 5;
A3: Following s,2 = Following (Following s) by Th15;
( (Following s) . [<*x,y*>,'&' ] = a1 '&' a2 & (Following s) . [<*y,c*>,'&' ] = a2 '&' a3 & (Following s) . [<*c,x*>,'&' ] = a3 '&' a1 ) by A1, Lm3;
hence (Following s,2) . (MajorityOutput x,y,c) = ((a1 '&' a2) 'or' (a2 '&' a3)) 'or' (a3 '&' a1) by A3, Th79; :: thesis: ( (Following s,2) . [<*x,y*>,'&' ] = a1 '&' a2 & (Following s,2) . [<*y,c*>,'&' ] = a2 '&' a3 & (Following s,2) . [<*c,x*>,'&' ] = a3 '&' a1 )
thus ( (Following s,2) . [<*x,y*>,'&' ] = a1 '&' a2 & (Following s,2) . [<*y,c*>,'&' ] = a2 '&' a3 & (Following s,2) . [<*c,x*>,'&' ] = a3 '&' a1 ) by A1, A2, A3, Lm3; :: thesis: verum