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 aC1C2a, aC2C3a, aC3C1a being Element of BOOLEAN st aC1C2a = s . [<*(GFA0CarryOutput (x1,x2,x3)),(GFA0CarryOutput (x5,x6,x7))*>,and2] & aC2C3a = s . [<*(GFA0CarryOutput (x5,x6,x7)),(GFA0CarryOutput ((GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7)),x4))*>,and2] & aC3C1a = s . [<*(GFA0CarryOutput ((GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7)),x4)),(GFA0CarryOutput (x1,x2,x3))*>,and2] holds
(Following s) . (GFA0CarryOutput ((GFA0CarryOutput (x1,x2,x3)),(GFA0CarryOutput (x5,x6,x7)),(GFA0CarryOutput ((GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7)),x4)))) = (aC1C2a 'or' aC2C3a) 'or' aC3C1a
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 aC1C2a, aC2C3a, aC3C1a being Element of BOOLEAN st aC1C2a = s . [<*(GFA0CarryOutput (x1,x2,x3)),(GFA0CarryOutput (x5,x6,x7))*>,and2] & aC2C3a = s . [<*(GFA0CarryOutput (x5,x6,x7)),(GFA0CarryOutput ((GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7)),x4))*>,and2] & aC3C1a = s . [<*(GFA0CarryOutput ((GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7)),x4)),(GFA0CarryOutput (x1,x2,x3))*>,and2] holds
(Following s) . (GFA0CarryOutput ((GFA0CarryOutput (x1,x2,x3)),(GFA0CarryOutput (x5,x6,x7)),(GFA0CarryOutput ((GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7)),x4)))) = (aC1C2a 'or' aC2C3a) 'or' aC3C1a
let aC1C2a, aC2C3a, aC3C1a be Element of BOOLEAN ; :: thesis: ( aC1C2a = s . [<*(GFA0CarryOutput (x1,x2,x3)),(GFA0CarryOutput (x5,x6,x7))*>,and2] & aC2C3a = s . [<*(GFA0CarryOutput (x5,x6,x7)),(GFA0CarryOutput ((GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7)),x4))*>,and2] & aC3C1a = s . [<*(GFA0CarryOutput ((GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7)),x4)),(GFA0CarryOutput (x1,x2,x3))*>,and2] implies (Following s) . (GFA0CarryOutput ((GFA0CarryOutput (x1,x2,x3)),(GFA0CarryOutput (x5,x6,x7)),(GFA0CarryOutput ((GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7)),x4)))) = (aC1C2a 'or' aC2C3a) 'or' aC3C1a )
assume A1: ( aC1C2a = s . [<*(GFA0CarryOutput (x1,x2,x3)),(GFA0CarryOutput (x5,x6,x7))*>,and2] & aC2C3a = s . [<*(GFA0CarryOutput (x5,x6,x7)),(GFA0CarryOutput ((GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7)),x4))*>,and2] & aC3C1a = s . [<*(GFA0CarryOutput ((GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7)),x4)),(GFA0CarryOutput (x1,x2,x3))*>,and2] ) ; :: thesis: (Following s) . (GFA0CarryOutput ((GFA0CarryOutput (x1,x2,x3)),(GFA0CarryOutput (x5,x6,x7)),(GFA0CarryOutput ((GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7)),x4)))) = (aC1C2a 'or' aC2C3a) 'or' aC3C1a
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))*>,and2] in the carrier of (STC0Str (x1,x2,x3,x4,x5,x6,x7)) & [<*(GFA0CarryOutput (x5,x6,x7)),(GFA0CarryOutput ((GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7)),x4))*>,and2] in the carrier of (STC0Str (x1,x2,x3,x4,x5,x6,x7)) & [<*(GFA0CarryOutput ((GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7)),x4)),(GFA0CarryOutput (x1,x2,x3))*>,and2] 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 ((GFA0CarryOutput (x1,x2,x3)),(GFA0CarryOutput (x5,x6,x7)),(GFA0CarryOutput ((GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7)),x4)))) = or3 . (s * <*[<*(GFA0CarryOutput (x1,x2,x3)),(GFA0CarryOutput (x5,x6,x7))*>,and2],[<*(GFA0CarryOutput (x5,x6,x7)),(GFA0CarryOutput ((GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7)),x4))*>,and2],[<*(GFA0CarryOutput ((GFA0AdderOutput (x1,x2,x3)),(GFA0AdderOutput (x5,x6,x7)),x4)),(GFA0CarryOutput (x1,x2,x3))*>,and2]*>) by ThSTC0S7, FACIRC_1:35
.= or3 . <*aC1C2a,aC2C3a,aC3C1a*> by A1, A2, A3, FINSEQ_2:126
.= (aC1C2a 'or' aC2C3a) 'or' aC3C1a by FACIRC_1:def 7 ;
:: thesis: verum