let x, y, c be object ; :: thesis: for f being Function of (2 -tuples_on BOOLEAN),BOOLEAN

for s being State of (2GatesCircuit (x,y,c,f)) st c <> [<*x,y*>,f] holds

( (Following s) . (2GatesCircOutput (x,y,c,f)) = f . <*(s . [<*x,y*>,f]),(s . c)*> & (Following s) . [<*x,y*>,f] = f . <*(s . x),(s . y)*> & (Following s) . x = s . x & (Following s) . y = s . y & (Following s) . c = s . c )

let f be Function of (2 -tuples_on BOOLEAN),BOOLEAN; :: thesis: for s being State of (2GatesCircuit (x,y,c,f)) st c <> [<*x,y*>,f] holds

( (Following s) . (2GatesCircOutput (x,y,c,f)) = f . <*(s . [<*x,y*>,f]),(s . c)*> & (Following s) . [<*x,y*>,f] = f . <*(s . x),(s . y)*> & (Following s) . x = s . x & (Following s) . y = s . y & (Following s) . c = s . c )

let s be State of (2GatesCircuit (x,y,c,f)); :: thesis: ( c <> [<*x,y*>,f] implies ( (Following s) . (2GatesCircOutput (x,y,c,f)) = f . <*(s . [<*x,y*>,f]),(s . c)*> & (Following s) . [<*x,y*>,f] = f . <*(s . x),(s . y)*> & (Following s) . x = s . x & (Following s) . y = s . y & (Following s) . c = s . c ) )

set S1 = 1GateCircStr (<*x,y*>,f);

set A1 = 1GateCircuit (x,y,f);

reconsider vx = x, vy = y as Vertex of (1GateCircStr (<*x,y*>,f)) by Th43;

reconsider s1 = s | the carrier of (1GateCircStr (<*x,y*>,f)) as State of (1GateCircuit (x,y,f)) by Th26;

set p = <*[<*x,y*>,f],c*>;

set xyf = [<*x,y*>,f];

set S2 = 1GateCircStr (<*[<*x,y*>,f],c*>,f);

set A2 = 1GateCircuit ([<*x,y*>,f],c,f);

set S = 2GatesCircStr (x,y,c,f);

A1: dom s1 = the carrier of (1GateCircStr (<*x,y*>,f)) by CIRCUIT1:3;

reconsider v2 = [<*[<*x,y*>,f],c*>,f] as Element of InnerVertices (1GateCircStr (<*[<*x,y*>,f],c*>,f)) by Th47;

InnerVertices (2GatesCircStr (x,y,c,f)) = {[<*x,y*>,f],(2GatesCircOutput (x,y,c,f))} by Th56;

then reconsider xyf = [<*x,y*>,f] as Element of InnerVertices (2GatesCircStr (x,y,c,f)) by TARSKI:def 2;

A2: rng <*[<*x,y*>,f],c*> = {xyf,c} by FINSEQ_2:127;

then c in rng <*[<*x,y*>,f],c*> by TARSKI:def 2;

then A3: c in InputVertices (1GateCircStr (<*[<*x,y*>,f],c*>,f)) by CIRCCOMB:42;

xyf in rng <*[<*x,y*>,f],c*> by A2, TARSKI:def 2;

then xyf in InputVertices (1GateCircStr (<*[<*x,y*>,f],c*>,f)) by CIRCCOMB:42;

then reconsider xyf9 = xyf, c9 = c as Vertex of (1GateCircStr (<*[<*x,y*>,f],c*>,f)) by A3;

reconsider xyf1 = xyf as Element of InnerVertices (1GateCircStr (<*x,y*>,f)) by Th47;

reconsider s2 = s | the carrier of (1GateCircStr (<*[<*x,y*>,f],c*>,f)) as State of (1GateCircuit ([<*x,y*>,f],c,f)) by Th26;

A4: dom s2 = the carrier of (1GateCircStr (<*[<*x,y*>,f],c*>,f)) by CIRCUIT1:3;

assume c <> [<*x,y*>,f] ; :: thesis: ( (Following s) . (2GatesCircOutput (x,y,c,f)) = f . <*(s . [<*x,y*>,f]),(s . c)*> & (Following s) . [<*x,y*>,f] = f . <*(s . x),(s . y)*> & (Following s) . x = s . x & (Following s) . y = s . y & (Following s) . c = s . c )

then A5: InputVertices (2GatesCircStr (x,y,c,f)) = {x,y,c} by Th57;

then A6: c in InputVertices (2GatesCircStr (x,y,c,f)) by ENUMSET1:def 1;

thus (Following s) . (2GatesCircOutput (x,y,c,f)) = (Following s2) . v2 by CIRCCOMB:64

.= f . <*(s2 . xyf9),(s2 . c9)*> by Th48

.= f . <*(s . [<*x,y*>,f]),(s2 . c9)*> by A4, FUNCT_1:47

.= f . <*(s . [<*x,y*>,f]),(s . c)*> by A4, FUNCT_1:47 ; :: thesis: ( (Following s) . [<*x,y*>,f] = f . <*(s . x),(s . y)*> & (Following s) . x = s . x & (Following s) . y = s . y & (Following s) . c = s . c )

thus (Following s) . [<*x,y*>,f] = (Following s1) . xyf1 by CIRCCOMB:64

.= f . <*(s1 . vx),(s1 . vy)*> by Th48

.= f . <*(s . x),(s1 . vy)*> by A1, FUNCT_1:47

.= f . <*(s . x),(s . y)*> by A1, FUNCT_1:47 ; :: thesis: ( (Following s) . x = s . x & (Following s) . y = s . y & (Following s) . c = s . c )

( x in InputVertices (2GatesCircStr (x,y,c,f)) & y in InputVertices (2GatesCircStr (x,y,c,f)) ) by A5, ENUMSET1:def 1;

hence ( (Following s) . x = s . x & (Following s) . y = s . y & (Following s) . c = s . c ) by A6, CIRCUIT2:def 5; :: thesis: verum

for s being State of (2GatesCircuit (x,y,c,f)) st c <> [<*x,y*>,f] holds

( (Following s) . (2GatesCircOutput (x,y,c,f)) = f . <*(s . [<*x,y*>,f]),(s . c)*> & (Following s) . [<*x,y*>,f] = f . <*(s . x),(s . y)*> & (Following s) . x = s . x & (Following s) . y = s . y & (Following s) . c = s . c )

let f be Function of (2 -tuples_on BOOLEAN),BOOLEAN; :: thesis: for s being State of (2GatesCircuit (x,y,c,f)) st c <> [<*x,y*>,f] holds

( (Following s) . (2GatesCircOutput (x,y,c,f)) = f . <*(s . [<*x,y*>,f]),(s . c)*> & (Following s) . [<*x,y*>,f] = f . <*(s . x),(s . y)*> & (Following s) . x = s . x & (Following s) . y = s . y & (Following s) . c = s . c )

let s be State of (2GatesCircuit (x,y,c,f)); :: thesis: ( c <> [<*x,y*>,f] implies ( (Following s) . (2GatesCircOutput (x,y,c,f)) = f . <*(s . [<*x,y*>,f]),(s . c)*> & (Following s) . [<*x,y*>,f] = f . <*(s . x),(s . y)*> & (Following s) . x = s . x & (Following s) . y = s . y & (Following s) . c = s . c ) )

set S1 = 1GateCircStr (<*x,y*>,f);

set A1 = 1GateCircuit (x,y,f);

reconsider vx = x, vy = y as Vertex of (1GateCircStr (<*x,y*>,f)) by Th43;

reconsider s1 = s | the carrier of (1GateCircStr (<*x,y*>,f)) as State of (1GateCircuit (x,y,f)) by Th26;

set p = <*[<*x,y*>,f],c*>;

set xyf = [<*x,y*>,f];

set S2 = 1GateCircStr (<*[<*x,y*>,f],c*>,f);

set A2 = 1GateCircuit ([<*x,y*>,f],c,f);

set S = 2GatesCircStr (x,y,c,f);

A1: dom s1 = the carrier of (1GateCircStr (<*x,y*>,f)) by CIRCUIT1:3;

reconsider v2 = [<*[<*x,y*>,f],c*>,f] as Element of InnerVertices (1GateCircStr (<*[<*x,y*>,f],c*>,f)) by Th47;

InnerVertices (2GatesCircStr (x,y,c,f)) = {[<*x,y*>,f],(2GatesCircOutput (x,y,c,f))} by Th56;

then reconsider xyf = [<*x,y*>,f] as Element of InnerVertices (2GatesCircStr (x,y,c,f)) by TARSKI:def 2;

A2: rng <*[<*x,y*>,f],c*> = {xyf,c} by FINSEQ_2:127;

then c in rng <*[<*x,y*>,f],c*> by TARSKI:def 2;

then A3: c in InputVertices (1GateCircStr (<*[<*x,y*>,f],c*>,f)) by CIRCCOMB:42;

xyf in rng <*[<*x,y*>,f],c*> by A2, TARSKI:def 2;

then xyf in InputVertices (1GateCircStr (<*[<*x,y*>,f],c*>,f)) by CIRCCOMB:42;

then reconsider xyf9 = xyf, c9 = c as Vertex of (1GateCircStr (<*[<*x,y*>,f],c*>,f)) by A3;

reconsider xyf1 = xyf as Element of InnerVertices (1GateCircStr (<*x,y*>,f)) by Th47;

reconsider s2 = s | the carrier of (1GateCircStr (<*[<*x,y*>,f],c*>,f)) as State of (1GateCircuit ([<*x,y*>,f],c,f)) by Th26;

A4: dom s2 = the carrier of (1GateCircStr (<*[<*x,y*>,f],c*>,f)) by CIRCUIT1:3;

assume c <> [<*x,y*>,f] ; :: thesis: ( (Following s) . (2GatesCircOutput (x,y,c,f)) = f . <*(s . [<*x,y*>,f]),(s . c)*> & (Following s) . [<*x,y*>,f] = f . <*(s . x),(s . y)*> & (Following s) . x = s . x & (Following s) . y = s . y & (Following s) . c = s . c )

then A5: InputVertices (2GatesCircStr (x,y,c,f)) = {x,y,c} by Th57;

then A6: c in InputVertices (2GatesCircStr (x,y,c,f)) by ENUMSET1:def 1;

thus (Following s) . (2GatesCircOutput (x,y,c,f)) = (Following s2) . v2 by CIRCCOMB:64

.= f . <*(s2 . xyf9),(s2 . c9)*> by Th48

.= f . <*(s . [<*x,y*>,f]),(s2 . c9)*> by A4, FUNCT_1:47

.= f . <*(s . [<*x,y*>,f]),(s . c)*> by A4, FUNCT_1:47 ; :: thesis: ( (Following s) . [<*x,y*>,f] = f . <*(s . x),(s . y)*> & (Following s) . x = s . x & (Following s) . y = s . y & (Following s) . c = s . c )

thus (Following s) . [<*x,y*>,f] = (Following s1) . xyf1 by CIRCCOMB:64

.= f . <*(s1 . vx),(s1 . vy)*> by Th48

.= f . <*(s . x),(s1 . vy)*> by A1, FUNCT_1:47

.= f . <*(s . x),(s . y)*> by A1, FUNCT_1:47 ; :: thesis: ( (Following s) . x = s . x & (Following s) . y = s . y & (Following s) . c = s . c )

( x in InputVertices (2GatesCircStr (x,y,c,f)) & y in InputVertices (2GatesCircStr (x,y,c,f)) ) by A5, ENUMSET1:def 1;

hence ( (Following s) . x = s . x & (Following s) . y = s . y & (Following s) . c = s . c ) by A6, CIRCUIT2:def 5; :: thesis: verum