let x, y, c be set ; :: thesis: for s being State of (BorrowCirc x,y,c)
for a1, a2, a3 being Element of BOOLEAN st a1 = s . x & a2 = s . y & a3 = s . c holds
( (Following s) . [<*x,y*>,and2a ] = ('not' a1) '&' a2 & (Following s) . [<*y,c*>,and2 ] = a2 '&' a3 & (Following s) . [<*x,c*>,and2a ] = ('not' a1) '&' a3 )
let s be State of (BorrowCirc x,y,c); :: thesis: for a1, a2, a3 being Element of BOOLEAN st a1 = s . x & a2 = s . y & a3 = s . c holds
( (Following s) . [<*x,y*>,and2a ] = ('not' a1) '&' a2 & (Following s) . [<*y,c*>,and2 ] = a2 '&' a3 & (Following s) . [<*x,c*>,and2a ] = ('not' a1) '&' a3 )
let a1, a2, a3 be Element of BOOLEAN ; :: thesis: ( a1 = s . x & a2 = s . y & a3 = s . c implies ( (Following s) . [<*x,y*>,and2a ] = ('not' a1) '&' a2 & (Following s) . [<*y,c*>,and2 ] = a2 '&' a3 & (Following s) . [<*x,c*>,and2a ] = ('not' a1) '&' a3 ) )
assume A1: ( a1 = s . x & a2 = s . y & a3 = s . c ) ; :: thesis: ( (Following s) . [<*x,y*>,and2a ] = ('not' a1) '&' a2 & (Following s) . [<*y,c*>,and2 ] = a2 '&' a3 & (Following s) . [<*x,c*>,and2a ] = ('not' a1) '&' a3 )
set S = BorrowStr x,y,c;
A2: InnerVertices (BorrowStr x,y,c) = the carrier' of (BorrowStr x,y,c) by FACIRC_1:37;
A3: dom s = the carrier of (BorrowStr x,y,c) by CIRCUIT1:4;
A4: ( x in the carrier of (BorrowStr x,y,c) & y in the carrier of (BorrowStr x,y,c) & c in the carrier of (BorrowStr x,y,c) ) by FSCIRC_1:6;
[<*x,y*>,and2a ] in InnerVertices (BorrowStr x,y,c) by FSCIRC_1:7;
hence (Following s) . [<*x,y*>,and2a ] = and2a . (s * <*x,y*>) by A2, FACIRC_1:35
.= and2a . <*a1,a2*> by A1, A3, A4, FINSEQ_2:145
.= ('not' a1) '&' a2 by TWOSCOMP:def 2 ;
:: thesis: ( (Following s) . [<*y,c*>,and2 ] = a2 '&' a3 & (Following s) . [<*x,c*>,and2a ] = ('not' a1) '&' a3 )
[<*y,c*>,and2 ] in InnerVertices (BorrowStr x,y,c) by FSCIRC_1:7;
hence (Following s) . [<*y,c*>,and2 ] = and2 . (s * <*y,c*>) by A2, FACIRC_1:35
.= and2 . <*a2,a3*> by A1, A3, A4, FINSEQ_2:145
.= a2 '&' a3 by TWOSCOMP:def 1 ;
:: thesis: (Following s) . [<*x,c*>,and2a ] = ('not' a1) '&' a3
[<*x,c*>,and2a ] in InnerVertices (BorrowStr x,y,c) by FSCIRC_1:7;
hence (Following s) . [<*x,c*>,and2a ] = and2a . (s * <*x,c*>) by A2, FACIRC_1:35
.= and2a . <*a1,a3*> by A1, A3, A4, FINSEQ_2:145
.= ('not' a1) '&' a3 by TWOSCOMP:def 2 ;
:: thesis: verum