let x, b be non pair set ; :: thesis: for s being State of (CompCirc x,b) holds
( (Following s) . (CompOutput x,b) = xor2a . <*(s . x),(s . b)*> & (Following s) . x = s . x & (Following s) . b = s . b )
let s be State of (CompCirc x,b); :: thesis: ( (Following s) . (CompOutput x,b) = xor2a . <*(s . x),(s . b)*> & (Following s) . x = s . x & (Following s) . b = s . b )
set p = <*x,b*>;
set S = CompStr x,b;
InputVertices (CompStr x,b) = {x,b} by FACIRC_1:40;
then A1: ( x in InputVertices (CompStr x,b) & b in InputVertices (CompStr x,b) ) by TARSKI:def 2;
A2: InnerVertices (CompStr x,b) = the carrier' of (CompStr x,b) by FACIRC_1:37;
A3: dom s = the carrier of (CompStr x,b) by CIRCUIT1:4;
A4: ( x in the carrier of (CompStr x,b) & b in the carrier of (CompStr x,b) ) by FACIRC_1:43;
thus (Following s) . (CompOutput x,b) = xor2a . (s * <*x,b*>) by A2, FACIRC_1:35
.= xor2a . <*(s . x),(s . b)*> by A3, A4, FINSEQ_2:145 ; :: thesis: ( (Following s) . x = s . x & (Following s) . b = s . b )
thus ( (Following s) . x = s . x & (Following s) . b = s . b ) by A1, CIRCUIT2:def 5; :: thesis: verum