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, aC3 being Element of BOOLEAN st aC1 = s . (GFA0CarryOutput (x1,x2,x3)) & aC2 = s . (GFA0CarryOutput (x5,x6,x7)) & aC3 = s . (GFA0CarryOutput ((GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7)),x4)) holds
( (Following s) . [<*(GFA0CarryOutput (x1,x2,x3)),(GFA0CarryOutput (x5,x6,x7))*>,and2] = aC1 '&' aC2 & (Following s) . [<*(GFA0CarryOutput (x5,x6,x7)),(GFA0CarryOutput ((GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7)),x4))*>,and2] = aC2 '&' aC3 & (Following s) . [<*(GFA0CarryOutput ((GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7)),x4)),(GFA0CarryOutput (x1,x2,x3))*>,and2] = aC3 '&' aC1 )
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 C1C2a = [<*(GFA0CarryOutput (x1,x2,x3)),(GFA0CarryOutput (x5,x6,x7))*>,and2];
set C2C3a = [<*(GFA0CarryOutput (x5,x6,x7)),(GFA0CarryOutput ((GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7)),x4))*>,and2];
set C3C1a = [<*(GFA0CarryOutput ((GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7)),x4)),(GFA0CarryOutput (x1,x2,x3))*>,and2];
let s be State of (STC0Circ (x1,x2,x3,x4,x5,x6,x7)); :: thesis: for aC1, aC2, aC3 being Element of BOOLEAN st aC1 = s . (GFA0CarryOutput (x1,x2,x3)) & aC2 = s . (GFA0CarryOutput (x5,x6,x7)) & aC3 = s . (GFA0CarryOutput ((GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7)),x4)) holds
( (Following s) . [<*(GFA0CarryOutput (x1,x2,x3)),(GFA0CarryOutput (x5,x6,x7))*>,and2] = aC1 '&' aC2 & (Following s) . [<*(GFA0CarryOutput (x5,x6,x7)),(GFA0CarryOutput ((GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7)),x4))*>,and2] = aC2 '&' aC3 & (Following s) . [<*(GFA0CarryOutput ((GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7)),x4)),(GFA0CarryOutput (x1,x2,x3))*>,and2] = aC3 '&' aC1 )
let aC1, aC2, aC3 be Element of BOOLEAN ; :: thesis: ( aC1 = s . (GFA0CarryOutput (x1,x2,x3)) & aC2 = s . (GFA0CarryOutput (x5,x6,x7)) & aC3 = s . (GFA0CarryOutput ((GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7)),x4)) implies ( (Following s) . [<*(GFA0CarryOutput (x1,x2,x3)),(GFA0CarryOutput (x5,x6,x7))*>,and2] = aC1 '&' aC2 & (Following s) . [<*(GFA0CarryOutput (x5,x6,x7)),(GFA0CarryOutput ((GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7)),x4))*>,and2] = aC2 '&' aC3 & (Following s) . [<*(GFA0CarryOutput ((GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7)),x4)),(GFA0CarryOutput (x1,x2,x3))*>,and2] = aC3 '&' aC1 ) )
assume A1: ( aC1 = s . (GFA0CarryOutput (x1,x2,x3)) & aC2 = s . (GFA0CarryOutput (x5,x6,x7)) & aC3 = s . (GFA0CarryOutput ((GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7)),x4)) ) ; :: thesis: ( (Following s) . [<*(GFA0CarryOutput (x1,x2,x3)),(GFA0CarryOutput (x5,x6,x7))*>,and2] = aC1 '&' aC2 & (Following s) . [<*(GFA0CarryOutput (x5,x6,x7)),(GFA0CarryOutput ((GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7)),x4))*>,and2] = aC2 '&' aC3 & (Following s) . [<*(GFA0CarryOutput ((GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7)),x4)),(GFA0CarryOutput (x1,x2,x3))*>,and2] = aC3 '&' aC1 )
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)) & GFA0CarryOutput ((GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7)),x4) in the carrier of (STC0Str (x1,x2,x3,x4,x5,x6,x7)) ) by ThSTC0S6;
A4: 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))*>,and2] = and2 . (s * <*(GFA0CarryOutput (x1,x2,x3)),(GFA0CarryOutput (x5,x6,x7))*>) by ThSTC0S7, FACIRC_1:35
.= and2 . <*aC1,aC2*> by A1, A2, A3, FINSEQ_2:125
.= aC1 '&' aC2 by FACIRC_1:def 6 ;
:: thesis: ( (Following s) . [<*(GFA0CarryOutput (x5,x6,x7)),(GFA0CarryOutput ((GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7)),x4))*>,and2] = aC2 '&' aC3 & (Following s) . [<*(GFA0CarryOutput ((GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7)),x4)),(GFA0CarryOutput (x1,x2,x3))*>,and2] = aC3 '&' aC1 )
A6: (Following s) . [<*(GFA0CarryOutput (x5,x6,x7)),(GFA0CarryOutput ((GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7)),x4))*>,and2] = and2 . (s * <*(GFA0CarryOutput (x5,x6,x7)),(GFA0CarryOutput ((GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7)),x4))*>) by A4, ThSTC0S7, FACIRC_1:35
.= and2 . <*aC2,aC3*> by A1, A2, A3, FINSEQ_2:125
.= aC2 '&' aC3 by FACIRC_1:def 6 ;
(Following s) . [<*(GFA0CarryOutput ((GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7)),x4)),(GFA0CarryOutput (x1,x2,x3))*>,and2] = and2 . (s * <*(GFA0CarryOutput ((GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7)),x4)),(GFA0CarryOutput (x1,x2,x3))*>) by A4, ThSTC0S7, FACIRC_1:35
.= and2 . <*aC3,aC1*> by A1, A2, A3, FINSEQ_2:125
.= aC3 '&' aC1 by FACIRC_1:def 6 ;
hence ( (Following s) . [<*(GFA0CarryOutput (x5,x6,x7)),(GFA0CarryOutput ((GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7)),x4))*>,and2] = aC2 '&' aC3 & (Following s) . [<*(GFA0CarryOutput ((GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7)),x4)),(GFA0CarryOutput (x1,x2,x3))*>,and2] = aC3 '&' aC1 ) by A6; :: thesis: verum