let x, y, z be set ; for s being State of (GFA0CarryCirc (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*>,and2] = a1 '&' a2 & (Following s) . [<*y,z*>,and2] = a2 '&' a3 & (Following s) . [<*z,x*>,and2] = a3 '&' a1 )
set f1 = and2 ;
set f2 = and2 ;
set f3 = and2 ;
let s be State of (GFA0CarryCirc (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*>,and2] = a1 '&' a2 & (Following s) . [<*y,z*>,and2] = a2 '&' a3 & (Following s) . [<*z,x*>,and2] = a3 '&' a1 )
set xy = [<*x,y*>,and2];
set yz = [<*y,z*>,and2];
set zx = [<*z,x*>,and2];
let a1, a2, a3 be Element of BOOLEAN ; ( a1 = s . x & a2 = s . y & a3 = s . z implies ( (Following s) . [<*x,y*>,and2] = a1 '&' a2 & (Following s) . [<*y,z*>,and2] = a2 '&' a3 & (Following s) . [<*z,x*>,and2] = a3 '&' a1 ) )
assume that
A1:
a1 = s . x
and
A2:
a2 = s . y
and
A3:
a3 = s . z
; ( (Following s) . [<*x,y*>,and2] = a1 '&' a2 & (Following s) . [<*y,z*>,and2] = a2 '&' a3 & (Following s) . [<*z,x*>,and2] = a3 '&' a1 )
set S = GFA0CarryStr (x,y,z);
A4:
InnerVertices (GFA0CarryStr (x,y,z)) = the carrier' of (GFA0CarryStr (x,y,z))
by FACIRC_1:37;
A5:
y in the carrier of (GFA0CarryStr (x,y,z))
by Th16;
A6:
x in the carrier of (GFA0CarryStr (x,y,z))
by Th16;
A7:
dom s = the carrier of (GFA0CarryStr (x,y,z))
by CIRCUIT1:3;
[<*x,y*>,and2] in InnerVertices (GFA0CarryStr (x,y,z))
by Th17;
hence (Following s) . [<*x,y*>,and2] =
and2 . (s * <*x,y*>)
by A4, FACIRC_1:35
.=
and2 . <*a1,a2*>
by A1, A2, A7, A6, A5, FINSEQ_2:125
.=
a1 '&' a2
by FACIRC_1:def 6
;
( (Following s) . [<*y,z*>,and2] = a2 '&' a3 & (Following s) . [<*z,x*>,and2] = a3 '&' a1 )
A8:
z in the carrier of (GFA0CarryStr (x,y,z))
by Th16;
[<*y,z*>,and2] in InnerVertices (GFA0CarryStr (x,y,z))
by Th17;
hence (Following s) . [<*y,z*>,and2] =
and2 . (s * <*y,z*>)
by A4, FACIRC_1:35
.=
and2 . <*a2,a3*>
by A2, A3, A7, A5, A8, FINSEQ_2:125
.=
a2 '&' a3
by FACIRC_1:def 6
;
(Following s) . [<*z,x*>,and2] = a3 '&' a1
[<*z,x*>,and2] in InnerVertices (GFA0CarryStr (x,y,z))
by Th17;
hence (Following s) . [<*z,x*>,and2] =
and2 . (s * <*z,x*>)
by A4, FACIRC_1:35
.=
and2 . <*a3,a1*>
by A1, A3, A7, A6, A8, FINSEQ_2:125
.=
a3 '&' a1
by FACIRC_1:def 6
;
verum