let x, y be set ; :: 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) . [<*x,y*>,f] = f . <*(s . x),(s . y)*> & (Following s) . x = s . x & (Following s) . y = s . y )
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) . [<*x,y*>,f] = f . <*(s . x),(s . y)*> & (Following s) . x = s . x & (Following s) . y = s . y )
let f be Function of (2 -tuples_on X),X; :: thesis: for s being State of (1GateCircuit <*x,y*>,f) holds
( (Following s) . [<*x,y*>,f] = f . <*(s . x),(s . y)*> & (Following s) . x = s . x & (Following s) . y = s . y )
let s be State of (1GateCircuit <*x,y*>,f); :: thesis: ( (Following s) . [<*x,y*>,f] = f . <*(s . x),(s . y)*> & (Following s) . x = s . x & (Following s) . y = s . y )
set p = <*x,y*>;
( x in {x,y} & y in {x,y} & dom s = the carrier of (1GateCircStr <*x,y*>,f) ) by CIRCUIT1:4, TARSKI:def 2;
then ( x in rng <*x,y*> & y in rng <*x,y*> & dom s = (rng <*x,y*>) \/ {[<*x,y*>,f]} ) by CIRCCOMB:def 6, FINSEQ_2:147;
then A1: ( x in dom s & y in dom s ) by XBOOLE_0:def 3;
thus (Following s) . [<*x,y*>,f] = f . (s * <*x,y*>) by CIRCCOMB:64
.= f . <*(s . x),(s . y)*> by A1, FINSEQ_2:145 ; :: thesis: ( (Following s) . x = s . x & (Following s) . y = s . y )
reconsider x = x, y = y as Vertex of (1GateCircStr <*x,y*>,f) by Th43;
InputVertices (1GateCircStr <*x,y*>,f) = rng <*x,y*> by CIRCCOMB:49
.= {x,y} by FINSEQ_2:147 ;
then ( x in InputVertices (1GateCircStr <*x,y*>,f) & y in InputVertices (1GateCircStr <*x,y*>,f) ) by TARSKI:def 2;
hence ( (Following s) . x = s . x & (Following s) . y = s . y ) by CIRCUIT2:def 5; :: thesis: verum