let x, y, c be set ; :: thesis: for s being State of (BorrowICirc x,y,c)
for a, b being Element of BOOLEAN st a = s . x & b = s . c holds
(Following s) . [<*x,c*>,and2a ] = ('not' a) '&' b
set xc = <*x,c*>;
set S3 = 1GateCircStr <*x,c*>,and2a ;
set A3 = 1GateCircuit x,c,and2a ;
set S = BorrowIStr x,y,c;
set A = BorrowICirc x,y,c;
set v3 = [<*x,c*>,and2a ];
let s be State of (BorrowICirc x,y,c); :: thesis: for a, b being Element of BOOLEAN st a = s . x & b = s . c holds
(Following s) . [<*x,c*>,and2a ] = ('not' a) '&' b
let a, b be Element of BOOLEAN ; :: thesis: ( a = s . x & b = s . c implies (Following s) . [<*x,c*>,and2a ] = ('not' a) '&' b )
assume A1: ( a = s . x & b = s . c ) ; :: thesis: (Following s) . [<*x,c*>,and2a ] = ('not' a) '&' b
reconsider xx = x, cc = c as Vertex of (1GateCircStr <*x,c*>,and2a ) by FACIRC_1:43;
reconsider s3 = s | the carrier of (1GateCircStr <*x,c*>,and2a ) as State of (1GateCircuit x,c,and2a ) by FACIRC_1:26;
reconsider v3 = [<*x,c*>,and2a ] as Element of InnerVertices (1GateCircStr <*x,c*>,and2a ) by FACIRC_1:47;
reconsider v = v3 as Element of InnerVertices (BorrowIStr x,y,c) by FACIRC_1:21;
A2: dom s3 = the carrier of (1GateCircStr <*x,c*>,and2a ) by CIRCUIT1:4;
reconsider xx = xx, cc = cc as Vertex of (BorrowIStr x,y,c) by FACIRC_1:20;
thus (Following s) . [<*x,c*>,and2a ] = (Following s3) . v by CIRCCOMB:72
.= and2a . <*(s3 . xx),(s3 . cc)*> by FACIRC_1:50
.= and2a . <*(s . xx),(s3 . cc)*> by A2, FUNCT_1:70
.= and2a . <*(s . xx),(s . cc)*> by A2, FUNCT_1:70
.= ('not' a) '&' b by A1, TWOSCOMP:def 2 ; :: thesis: verum