let x, y, z be set ; for s being State of (GFA3CarryCirc (x,y,z))
for a1, a2, a3 being Element of BOOLEAN st a1 = s . x & a2 = s . y & a3 = s . z holds
( (Following s) . [<*x,y*>,nor2] = ('not' a1) '&' ('not' a2) & (Following s) . [<*y,z*>,nor2] = ('not' a2) '&' ('not' a3) & (Following s) . [<*z,x*>,nor2] = ('not' a3) '&' ('not' a1) )
set f1 = nor2 ;
set f2 = nor2 ;
set f3 = nor2 ;
let s be State of (GFA3CarryCirc (x,y,z)); for a1, a2, a3 being Element of BOOLEAN st a1 = s . x & a2 = s . y & a3 = s . z holds
( (Following s) . [<*x,y*>,nor2] = ('not' a1) '&' ('not' a2) & (Following s) . [<*y,z*>,nor2] = ('not' a2) '&' ('not' a3) & (Following s) . [<*z,x*>,nor2] = ('not' a3) '&' ('not' a1) )
set xy = [<*x,y*>,nor2];
set yz = [<*y,z*>,nor2];
set zx = [<*z,x*>,nor2];
let a1, a2, a3 be Element of BOOLEAN ; ( a1 = s . x & a2 = s . y & a3 = s . z implies ( (Following s) . [<*x,y*>,nor2] = ('not' a1) '&' ('not' a2) & (Following s) . [<*y,z*>,nor2] = ('not' a2) '&' ('not' a3) & (Following s) . [<*z,x*>,nor2] = ('not' a3) '&' ('not' a1) ) )
assume that
A1:
a1 = s . x
and
A2:
a2 = s . y
and
A3:
a3 = s . z
; ( (Following s) . [<*x,y*>,nor2] = ('not' a1) '&' ('not' a2) & (Following s) . [<*y,z*>,nor2] = ('not' a2) '&' ('not' a3) & (Following s) . [<*z,x*>,nor2] = ('not' a3) '&' ('not' a1) )
set S = GFA3CarryStr (x,y,z);
A4:
InnerVertices (GFA3CarryStr (x,y,z)) = the carrier' of (GFA3CarryStr (x,y,z))
by FACIRC_1:37;
A5:
y in the carrier of (GFA3CarryStr (x,y,z))
by Th111;
A6:
x in the carrier of (GFA3CarryStr (x,y,z))
by Th111;
A7:
dom s = the carrier of (GFA3CarryStr (x,y,z))
by CIRCUIT1:3;
[<*x,y*>,nor2] in InnerVertices (GFA3CarryStr (x,y,z))
by Th112;
hence (Following s) . [<*x,y*>,nor2] =
nor2 . (s * <*x,y*>)
by A4, FACIRC_1:35
.=
nor2 . <*a1,a2*>
by A1, A2, A7, A6, A5, FINSEQ_2:125
.=
('not' a1) '&' ('not' a2)
by TWOSCOMP:54
;
( (Following s) . [<*y,z*>,nor2] = ('not' a2) '&' ('not' a3) & (Following s) . [<*z,x*>,nor2] = ('not' a3) '&' ('not' a1) )
A8:
z in the carrier of (GFA3CarryStr (x,y,z))
by Th111;
[<*y,z*>,nor2] in InnerVertices (GFA3CarryStr (x,y,z))
by Th112;
hence (Following s) . [<*y,z*>,nor2] =
nor2 . (s * <*y,z*>)
by A4, FACIRC_1:35
.=
nor2 . <*a2,a3*>
by A2, A3, A7, A5, A8, FINSEQ_2:125
.=
('not' a2) '&' ('not' a3)
by TWOSCOMP:54
;
(Following s) . [<*z,x*>,nor2] = ('not' a3) '&' ('not' a1)
[<*z,x*>,nor2] in InnerVertices (GFA3CarryStr (x,y,z))
by Th112;
hence (Following s) . [<*z,x*>,nor2] =
nor2 . (s * <*z,x*>)
by A4, FACIRC_1:35
.=
nor2 . <*a3,a1*>
by A1, A3, A7, A6, A8, FINSEQ_2:125
.=
('not' a3) '&' ('not' a1)
by TWOSCOMP:54
;
verum