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) . [<*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*>;

dom s = the carrier of (1GateCircStr (<*x,y*>,f)) by CIRCUIT1:3;

then A1: dom s = (rng <*x,y*>) \/ {[<*x,y*>,f]} by CIRCCOMB:def 6;

y in {x,y} by TARSKI:def 2;

then y in rng <*x,y*> by FINSEQ_2:127;

then A2: y in dom s by A1, XBOOLE_0:def 3;

x in {x,y} by TARSKI:def 2;

then x in rng <*x,y*> by FINSEQ_2:127;

then A3: x in dom s by A1, XBOOLE_0:def 3;

thus (Following s) . [<*x,y*>,f] = f . (s * <*x,y*>) by CIRCCOMB:56

.= f . <*(s . x),(s . y)*> by A3, A2, FINSEQ_2:125 ; :: 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:42

.= {x,y} by FINSEQ_2:127 ;

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

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*>;

dom s = the carrier of (1GateCircStr (<*x,y*>,f)) by CIRCUIT1:3;

then A1: dom s = (rng <*x,y*>) \/ {[<*x,y*>,f]} by CIRCCOMB:def 6;

y in {x,y} by TARSKI:def 2;

then y in rng <*x,y*> by FINSEQ_2:127;

then A2: y in dom s by A1, XBOOLE_0:def 3;

x in {x,y} by TARSKI:def 2;

then x in rng <*x,y*> by FINSEQ_2:127;

then A3: x in dom s by A1, XBOOLE_0:def 3;

thus (Following s) . [<*x,y*>,f] = f . (s * <*x,y*>) by CIRCCOMB:56

.= f . <*(s . x),(s . y)*> by A3, A2, FINSEQ_2:125 ; :: 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:42

.= {x,y} by FINSEQ_2:127 ;

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