let x, b be non pair set ; :: thesis: for s being State of (BitCompCirc (x,b)) holds
( (Following s) . (IncrementOutput (x,b)) = and2a . <*(s . x),(s . b)*> & (Following s) . x = s . x & (Following s) . b = s . b )
let s be State of (BitCompCirc (x,b)); :: thesis: ( (Following s) . (IncrementOutput (x,b)) = and2a . <*(s . x),(s . b)*> & (Following s) . x = s . x & (Following s) . b = s . b )
set p = <*x,b*>;
set S = BitCompStr (x,b);
A1: ( dom s = the carrier of (BitCompStr (x,b)) & x in the carrier of (BitCompStr (x,b)) ) by Th36, CIRCUIT1:3;
A2: b in the carrier of (BitCompStr (x,b)) by Th36;
InnerVertices (BitCompStr (x,b)) = the carrier' of (BitCompStr (x,b)) by FACIRC_1:37;
hence (Following s) . (IncrementOutput (x,b)) = and2a . (s * <*x,b*>) by Th39, FACIRC_1:35
.= and2a . <*(s . x),(s . b)*> by A1, A2, FINSEQ_2:125 ;
:: thesis: ( (Following s) . x = s . x & (Following s) . b = s . b )
InputVertices (BitCompStr (x,b)) = {x,b} by Th40;
then ( x in InputVertices (BitCompStr (x,b)) & b in InputVertices (BitCompStr (x,b)) ) by TARSKI:def 2;
hence ( (Following s) . x = s . x & (Following s) . b = s . b ) by CIRCUIT2:def 5; :: thesis: verum