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 aC1, aC2 being Element of BOOLEAN st aC1 = s . (GFA0CarryOutput (x1,x2,x3)) & aC2 = s . (GFA0CarryOutput (x5,x6,x7)) holds
(Following s) . [<*(GFA0CarryOutput (x1,x2,x3)),(GFA0CarryOutput (x5,x6,x7))*>,xor2] = aC1 'xor' aC2
set S = STC0Str (x1,x2,x3,x4,x5,x6,x7);
set C = STC0Circ (x1,x2,x3,x4,x5,x6,x7);
set C1out = GFA0CarryOutput (x1,x2,x3);
set C2out = GFA0CarryOutput (x5,x6,x7);
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 aC1, aC2 being Element of BOOLEAN st aC1 = s . (GFA0CarryOutput (x1,x2,x3)) & aC2 = s . (GFA0CarryOutput (x5,x6,x7)) holds
(Following s) . [<*(GFA0CarryOutput (x1,x2,x3)),(GFA0CarryOutput (x5,x6,x7))*>,xor2] = aC1 'xor' aC2
let aC1, aC2 be Element of BOOLEAN ; :: thesis: ( aC1 = s . (GFA0CarryOutput (x1,x2,x3)) & aC2 = s . (GFA0CarryOutput (x5,x6,x7)) implies (Following s) . [<*(GFA0CarryOutput (x1,x2,x3)),(GFA0CarryOutput (x5,x6,x7))*>,xor2] = aC1 'xor' aC2 )
assume A1: ( aC1 = s . (GFA0CarryOutput (x1,x2,x3)) & aC2 = s . (GFA0CarryOutput (x5,x6,x7)) ) ; :: thesis: (Following s) . [<*(GFA0CarryOutput (x1,x2,x3)),(GFA0CarryOutput (x5,x6,x7))*>,xor2] = aC1 'xor' aC2
A2: dom s = the carrier of (STC0Str (x1,x2,x3,x4,x5,x6,x7)) by CIRCUIT1:3;
A3: ( GFA0CarryOutput (x1,x2,x3) in the carrier of (STC0Str (x1,x2,x3,x4,x5,x6,x7)) & GFA0CarryOutput (x5,x6,x7) 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) . [<*(GFA0CarryOutput (x1,x2,x3)),(GFA0CarryOutput (x5,x6,x7))*>,xor2] = xor2 . (s * <*(GFA0CarryOutput (x1,x2,x3)),(GFA0CarryOutput (x5,x6,x7))*>) by ThSTC0S7, FACIRC_1:35
.= xor2 . <*aC1,aC2*> by A1, A2, A3, FINSEQ_2:125
.= aC1 'xor' aC2 by FACIRC_1:def 4 ;
:: thesis: verum