let x, y be object ; :: thesis: for X being non empty finite set
for f being Function of (2 -tuples_on X),X
for s being State of (1GateCircuit (<*x,y*>,f)) holds Following s is stable
let X be non empty finite set ; :: thesis: for f being Function of (2 -tuples_on X),X
for s being State of (1GateCircuit (<*x,y*>,f)) holds Following s is stable
let f be Function of (2 -tuples_on X),X; :: thesis: for s being State of (1GateCircuit (<*x,y*>,f)) holds Following s is stable
set S = 1GateCircStr (<*x,y*>,f);
let s be State of (1GateCircuit (<*x,y*>,f)); :: thesis: Following s is stable
set s1 = Following s;
set s2 = Following (Following s);
set p = <*x,y*>;
A1: the carrier of (1GateCircStr (<*x,y*>,f)) = (rng <*x,y*>) \/ {[<*x,y*>,f]} by CIRCCOMB:def 6
.= {x,y} \/ {[<*x,y*>,f]} by FINSEQ_2:127 ;

A2: now :: thesis: for a being object st a in the carrier of (1GateCircStr (<*x,y*>,f)) holds
(Following (Following s)) . a = (Following s) . a

let a be object ; :: thesis: ( a in the carrier of (1GateCircStr (<*x,y*>,f)) implies (Following (Following s)) . a = (Following s) . a )
A3: (Following (Following s)) . [<*x,y*>,f] = f . <*((Following s) . x),((Following s) . y)*> by Th48;
assume a in the carrier of (1GateCircStr (<*x,y*>,f)) ; :: thesis: (Following (Following s)) . a = (Following s) . a
then ( a in {x,y} or a in {[<*x,y*>,f]} ) by A1, XBOOLE_0:def 3;
then A4: ( a = x or a = y or a = [<*x,y*>,f] ) by TARSKI:def 1, TARSKI:def 2;
( (Following s) . x = s . x & (Following s) . y = s . y ) by Th48;
hence (Following (Following s)) . a = (Following s) . a by A4, A3, Th48; :: thesis: verum

end;

( dom (Following s) = the carrier of (1GateCircStr (<*x,y*>,f)) & dom (Following (Following s)) = the carrier of (1GateCircStr (<*x,y*>,f)) ) by CIRCUIT1:3;
hence Following s = Following (Following s) by A2, FUNCT_1:2; :: according to CIRCUIT2:def 6 :: thesis: verum