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
( () . [<*x,y*>,and2a] = ('not' a1) '&' a2 & () . [<*y,c*>,and2] = a2 '&' a3 & () . [<*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
( () . [<*x,y*>,and2a] = ('not' a1) '&' a2 & () . [<*y,c*>,and2] = a2 '&' a3 & () . [<*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 ( () . [<*x,y*>,and2a] = ('not' a1) '&' a2 & () . [<*y,c*>,and2] = a2 '&' a3 & () . [<*x,c*>,and2a] = ('not' a1) '&' a3 ) )
assume that
A1: a1 = s . x and
A2: a2 = s . y and
A3: a3 = s . c ; :: thesis: ( () . [<*x,y*>,and2a] = ('not' a1) '&' a2 & () . [<*y,c*>,and2] = a2 '&' a3 & () . [<*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 () . [<*x,y*>,and2a] = and2a . (s * <*x,y*>) by
.= and2a . <*a1,a2*> by
.= ('not' a1) '&' a2 by TWOSCOMP:def 2 ;
:: thesis: ( () . [<*y,c*>,and2] = a2 '&' a3 & () . [<*x,c*>,and2a] = ('not' a1) '&' a3 )
[<*y,c*>,and2] in InnerVertices (BorrowStr (x,y,c)) by FSCIRC_1:7;
hence () . [<*y,c*>,and2] = and2 . (s * <*y,c*>) by
.= and2 . <*a2,a3*> by
.= a2 '&' a3 by FACIRC_1:def 6 ;
:: thesis: () . [<*x,c*>,and2a] = ('not' a1) '&' a3
[<*x,c*>,and2a] in InnerVertices (BorrowStr (x,y,c)) by FSCIRC_1:7;
hence () . [<*x,c*>,and2a] = and2a . (s * <*x,c*>) by
.= and2a . <*a1,a3*> by
.= ('not' a1) '&' a3 by TWOSCOMP:def 2 ;
:: thesis: verum