let x, y, z be set ; :: thesis: for s being State of (GFA1CarryCirc x,y,z)
for a1, a2, a3 being Element of BOOLEAN st a1 = s . [<*x,y*>,and2c ] & a2 = s . [<*y,z*>,and2a ] & a3 = s . [<*z,x*>,and2 ] holds
(Following s) . (GFA1CarryOutput x,y,z) = (a1 'or' a2) 'or' a3
set f1 = and2c ;
set f2 = and2a ;
set f3 = and2 ;
set f4 = or3 ;
let s be State of (GFA1CarryCirc x,y,z); :: thesis: for a1, a2, a3 being Element of BOOLEAN st a1 = s . [<*x,y*>,and2c ] & a2 = s . [<*y,z*>,and2a ] & a3 = s . [<*z,x*>,and2 ] holds
(Following s) . (GFA1CarryOutput x,y,z) = (a1 'or' a2) 'or' a3
set xy = [<*x,y*>,and2c ];
set yz = [<*y,z*>,and2a ];
set zx = [<*z,x*>,and2 ];
let a1, a2, a3 be Element of BOOLEAN ; :: thesis: ( a1 = s . [<*x,y*>,and2c ] & a2 = s . [<*y,z*>,and2a ] & a3 = s . [<*z,x*>,and2 ] implies (Following s) . (GFA1CarryOutput x,y,z) = (a1 'or' a2) 'or' a3 )
assume A1: ( a1 = s . [<*x,y*>,and2c ] & a2 = s . [<*y,z*>,and2a ] & a3 = s . [<*z,x*>,and2 ] ) ; :: thesis: (Following s) . (GFA1CarryOutput x,y,z) = (a1 'or' a2) 'or' a3
set S = GFA1CarryStr x,y,z;
reconsider xy = [<*x,y*>,and2c ], yz = [<*y,z*>,and2a ], zx = [<*z,x*>,and2 ] as Element of InnerVertices (GFA1CarryStr x,y,z) by Th56;
A2: dom s = the carrier of (GFA1CarryStr x,y,z) by CIRCUIT1:4;
InnerVertices (GFA1CarryStr x,y,z) = the carrier' of (GFA1CarryStr x,y,z) by FACIRC_1:37;
hence (Following s) . (GFA1CarryOutput x,y,z) = or3 . (s * <*xy,yz,zx*>) by FACIRC_1:35
.= or3 . <*a1,a2,a3*> by A1, A2, FINSEQ_2:146
.= (a1 'or' a2) 'or' a3 by TWOSCOMP:16 ;
:: thesis: verum