let x1, x2, x3, x5, x6, x7 be non pair set ; :: thesis: for x4 being set
for s being State of (STC0ICirc (x1,x2,x3,x4,x5,x6,x7))
for a123567, a4 being Element of BOOLEAN st a123567 = s . [<*(GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7))*>,xor2] & a4 = s . x4 holds
(Following s) . (GFA0AdderOutput ((GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7)),x4)) = a123567 'xor' a4
let x4 be set ; :: thesis: for s being State of (STC0ICirc (x1,x2,x3,x4,x5,x6,x7))
for a123567, a4 being Element of BOOLEAN st a123567 = s . [<*(GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7))*>,xor2] & a4 = s . x4 holds
(Following s) . (GFA0AdderOutput ((GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7)),x4)) = a123567 'xor' a4
set S = STC0IStr (x1,x2,x3,x4,x5,x6,x7);
set C = STC0ICirc (x1,x2,x3,x4,x5,x6,x7);
set A1out = GFA0AdderOutput (x1,x2,x3);
set A2out = GFA0AdderOutput (x5,x6,x7);
set A1A20 = [<*(GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7))*>,xor2];
let s be State of (STC0ICirc (x1,x2,x3,x4,x5,x6,x7)); :: thesis: for a123567, a4 being Element of BOOLEAN st a123567 = s . [<*(GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7))*>,xor2] & a4 = s . x4 holds
(Following s) . (GFA0AdderOutput ((GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7)),x4)) = a123567 'xor' a4
let a123567, a4 be Element of BOOLEAN ; :: thesis: ( a123567 = s . [<*(GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7))*>,xor2] & a4 = s . x4 implies (Following s) . (GFA0AdderOutput ((GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7)),x4)) = a123567 'xor' a4 )
assume A1: ( a123567 = s . [<*(GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7))*>,xor2] & a4 = s . x4 ) ; :: thesis: (Following s) . (GFA0AdderOutput ((GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7)),x4)) = a123567 'xor' a4
A2: dom s = the carrier of (STC0IStr (x1,x2,x3,x4,x5,x6,x7)) by CIRCUIT1:3;
A3: ( [<*(GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7))*>,xor2] in the carrier of (STC0IStr (x1,x2,x3,x4,x5,x6,x7)) & x4 in the carrier of (STC0IStr (x1,x2,x3,x4,x5,x6,x7)) ) by ThSTC0IS6;
InnerVertices (STC0IStr (x1,x2,x3,x4,x5,x6,x7)) = the carrier' of (STC0IStr (x1,x2,x3,x4,x5,x6,x7)) by FACIRC_1:37;
hence (Following s) . (GFA0AdderOutput ((GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7)),x4)) = xor2 . (s * <*[<*(GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7))*>,xor2],x4*>) by ThSTC0IS7, FACIRC_1:35
.= xor2 . <*a123567,a4*> by A1, A2, A3, FINSEQ_2:125
.= a123567 'xor' a4 by FACIRC_1:def 4 ;
:: thesis: verum