let x1, x2, x3, x4, x5, x6, x7 be non pair set ; :: thesis: for s being State of (STC0Circ (x1,x2,x3,x4,x5,x6,x7))
for aC1C2x, aC3 being Element of BOOLEAN st aC1C2x = s . [<*(GFA0CarryOutput (x1,x2,x3)),(GFA0CarryOutput (x5,x6,x7))*>,xor2] & aC3 = s . (GFA0CarryOutput ((GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7)),x4)) holds
(Following s) . (GFA0AdderOutput ((GFA0CarryOutput (x1,x2,x3)),(GFA0CarryOutput (x5,x6,x7)),(GFA0CarryOutput ((GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7)),x4)))) = aC1C2x 'xor' aC3
set S = STC0Str (x1,x2,x3,x4,x5,x6,x7);
set C = STC0Circ (x1,x2,x3,x4,x5,x6,x7);
set A1out = GFA0AdderOutput (x1,x2,x3);
set A2out = GFA0AdderOutput (x5,x6,x7);
set C1out = GFA0CarryOutput (x1,x2,x3);
set C2out = GFA0CarryOutput (x5,x6,x7);
set C3out = GFA0CarryOutput ((GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7)),x4);
set C1C2x = [<*(GFA0CarryOutput (x1,x2,x3)),(GFA0CarryOutput (x5,x6,x7))*>,xor2];
let s be State of (STC0Circ (x1,x2,x3,x4,x5,x6,x7)); :: thesis: for aC1C2x, aC3 being Element of BOOLEAN st aC1C2x = s . [<*(GFA0CarryOutput (x1,x2,x3)),(GFA0CarryOutput (x5,x6,x7))*>,xor2] & aC3 = s . (GFA0CarryOutput ((GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7)),x4)) holds
(Following s) . (GFA0AdderOutput ((GFA0CarryOutput (x1,x2,x3)),(GFA0CarryOutput (x5,x6,x7)),(GFA0CarryOutput ((GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7)),x4)))) = aC1C2x 'xor' aC3
let aC1C2x, aC3 be Element of BOOLEAN ; :: thesis: ( aC1C2x = s . [<*(GFA0CarryOutput (x1,x2,x3)),(GFA0CarryOutput (x5,x6,x7))*>,xor2] & aC3 = s . (GFA0CarryOutput ((GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7)),x4)) implies (Following s) . (GFA0AdderOutput ((GFA0CarryOutput (x1,x2,x3)),(GFA0CarryOutput (x5,x6,x7)),(GFA0CarryOutput ((GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7)),x4)))) = aC1C2x 'xor' aC3 )
assume A1: ( aC1C2x = s . [<*(GFA0CarryOutput (x1,x2,x3)),(GFA0CarryOutput (x5,x6,x7))*>,xor2] & aC3 = s . (GFA0CarryOutput ((GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7)),x4)) ) ; :: thesis: (Following s) . (GFA0AdderOutput ((GFA0CarryOutput (x1,x2,x3)),(GFA0CarryOutput (x5,x6,x7)),(GFA0CarryOutput ((GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7)),x4)))) = aC1C2x 'xor' aC3
A2: dom s = the carrier of (STC0Str (x1,x2,x3,x4,x5,x6,x7)) by CIRCUIT1:3;
A3: ( [<*(GFA0CarryOutput (x1,x2,x3)),(GFA0CarryOutput (x5,x6,x7))*>,xor2] in the carrier of (STC0Str (x1,x2,x3,x4,x5,x6,x7)) & GFA0CarryOutput ((GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7)),x4) in the carrier of (STC0Str (x1,x2,x3,x4,x5,x6,x7)) ) by ThSTC0S6;
InnerVertices (STC0Str (x1,x2,x3,x4,x5,x6,x7)) = the carrier' of (STC0Str (x1,x2,x3,x4,x5,x6,x7)) by FACIRC_1:37;
hence (Following s) . (GFA0AdderOutput ((GFA0CarryOutput (x1,x2,x3)),(GFA0CarryOutput (x5,x6,x7)),(GFA0CarryOutput ((GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7)),x4)))) = xor2 . (s * <*[<*(GFA0CarryOutput (x1,x2,x3)),(GFA0CarryOutput (x5,x6,x7))*>,xor2],(GFA0CarryOutput ((GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7)),x4))*>) by ThSTC0S7, FACIRC_1:35
.= xor2 . <*aC1C2x,aC3*> by A1, A2, A3, FINSEQ_2:125
.= aC1C2x 'xor' aC3 by FACIRC_1:def 4 ;
:: thesis: verum