let f be Function of (2 -tuples_on BOOLEAN),BOOLEAN; :: thesis: for x, y, c being set st c <> [<*x,y*>,f] holds
for s being State of (2GatesCircuit (x,y,c,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 x, y, c be set ; :: thesis: ( c <> [<*x,y*>,f] implies for s being State of (2GatesCircuit (x,y,c,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 ) )
assume A1: c <> [<*x,y*>,f] ; :: thesis: for s being State of (2GatesCircuit (x,y,c,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 )
set S = 2GatesCircStr (x,y,c,f);
A2: InputVertices (2GatesCircStr (x,y,c,f)) = {x,y,c} by A1, FACIRC_1:57;
then A3: c in InputVertices (2GatesCircStr (x,y,c,f)) by ENUMSET1:def 1;
set xyf = [<*x,y*>,f];
set p = <*[<*x,y*>,f],c*>;
set S2 = 1GateCircStr (<*[<*x,y*>,f],c*>,f);
set A2 = 1GateCircuit ([<*x,y*>,f],c,f);
InnerVertices (2GatesCircStr (x,y,c,f)) = {[<*x,y*>,f],(2GatesCircOutput (x,y,c,f))} by FACIRC_1:56;
then reconsider xyf = [<*x,y*>,f] as Element of InnerVertices (2GatesCircStr (x,y,c,f)) by TARSKI:def 2;
set A = 2GatesCircuit (x,y,c,f);
let s be State of (2GatesCircuit (x,y,c,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 )
set S1 = 1GateCircStr (<*x,y*>,f);
set A1 = 1GateCircuit (x,y,f);
reconsider s1 = s | the carrier of (1GateCircStr (<*x,y*>,f)) as State of (1GateCircuit (x,y,f)) by FACIRC_1:26;
reconsider s2 = s | the carrier of (1GateCircStr (<*[<*x,y*>,f],c*>,f)) as State of (1GateCircuit ([<*x,y*>,f],c,f)) by FACIRC_1:26;
A4: dom s2 = the carrier of (1GateCircStr (<*[<*x,y*>,f],c*>,f)) by CIRCUIT1:3;
A5: rng <*[<*x,y*>,f],c*> = {xyf,c} by FINSEQ_2:127;
then c in rng <*[<*x,y*>,f],c*> by TARSKI:def 2;
then A6: c in InputVertices (1GateCircStr (<*[<*x,y*>,f],c*>,f)) by CIRCCOMB:42;
xyf in rng <*[<*x,y*>,f],c*> by A5, 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 A6;
reconsider v2 = [<*[<*x,y*>,f],c*>,f] as Element of InnerVertices (1GateCircStr (<*[<*x,y*>,f],c*>,f)) by FACIRC_1:47;
reconsider xyf1 = xyf as Element of InnerVertices (1GateCircStr (<*x,y*>,f)) by FACIRC_1:47;
reconsider vx = x, vy = y as Vertex of (1GateCircStr (<*x,y*>,f)) by FACIRC_1:43;
A7: dom s1 = the carrier of (1GateCircStr (<*x,y*>,f)) by CIRCUIT1:3;
thus (Following s) . (2GatesCircOutput (x,y,c,f)) = (Following s2) . v2 by CIRCCOMB:64
.= f . <*(s2 . xyf9),(s2 . c9)*> by FACIRC_1:50
.= 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 FACIRC_1:50
.= f . <*(s . x),(s1 . vy)*> by A7, FUNCT_1:47
.= f . <*(s . x),(s . y)*> by A7, 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 A2, ENUMSET1:def 1;
hence ( (Following s) . x = s . x & (Following s) . y = s . y & (Following s) . c = s . c ) by A3, CIRCUIT2:def 5; :: thesis: verum