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 that

A1: a1 = s . x and

A2: a2 = s . y and

A3: 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);

A4: InnerVertices (BorrowStr (x,y,c)) = the carrier' of (BorrowStr (x,y,c)) by FACIRC_1:37;

A5: dom s = the carrier of (BorrowStr (x,y,c)) by CIRCUIT1:3;

A6: x in the carrier of (BorrowStr (x,y,c)) by FSCIRC_1:6;

A7: y in the carrier of (BorrowStr (x,y,c)) by FSCIRC_1:6;

A8: 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 A4, FACIRC_1:35

.= and2a . <*a1,a2*> by A1, A2, A5, A6, A7, FINSEQ_2:125

.= ('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 A4, FACIRC_1:35

.= and2 . <*a2,a3*> by A2, A3, A5, A7, A8, FINSEQ_2:125

.= a2 '&' a3 by FACIRC_1:def 6 ;

:: 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 A4, FACIRC_1:35

.= and2a . <*a1,a3*> by A1, A3, A5, A6, A8, FINSEQ_2:125

.= ('not' a1) '&' a3 by TWOSCOMP:def 2 ;

:: thesis: verum

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 that

A1: a1 = s . x and

A2: a2 = s . y and

A3: 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);

A4: InnerVertices (BorrowStr (x,y,c)) = the carrier' of (BorrowStr (x,y,c)) by FACIRC_1:37;

A5: dom s = the carrier of (BorrowStr (x,y,c)) by CIRCUIT1:3;

A6: x in the carrier of (BorrowStr (x,y,c)) by FSCIRC_1:6;

A7: y in the carrier of (BorrowStr (x,y,c)) by FSCIRC_1:6;

A8: 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 A4, FACIRC_1:35

.= and2a . <*a1,a2*> by A1, A2, A5, A6, A7, FINSEQ_2:125

.= ('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 A4, FACIRC_1:35

.= and2 . <*a2,a3*> by A2, A3, A5, A7, A8, FINSEQ_2:125

.= a2 '&' a3 by FACIRC_1:def 6 ;

:: 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 A4, FACIRC_1:35

.= and2a . <*a1,a3*> by A1, A3, A5, A6, A8, FINSEQ_2:125

.= ('not' a1) '&' a3 by TWOSCOMP:def 2 ;

:: thesis: verum