let s be State of (F₉() +* (1GateCircuit <*F₁(),F₂(),F₃(),F₄(),F₅()*>,F₁₀())); :: thesis: for s' being State of F₉() st s' = s | the carrier of F₈() holds
for a1, a2, a3, a4, a5 being Element of F₆() st ( F₁() in InnerVertices F₈() implies a1 = (Result s') . F₁() ) & ( not F₁() in InnerVertices F₈() implies a1 = s . F₁() ) & ( F₂() in InnerVertices F₈() implies a2 = (Result s') . F₂() ) & ( not F₂() in InnerVertices F₈() implies a2 = s . F₂() ) & ( F₃() in InnerVertices F₈() implies a3 = (Result s') . F₃() ) & ( not F₃() in InnerVertices F₈() implies a3 = s . F₃() ) & ( F₄() in InnerVertices F₈() implies a4 = (Result s') . F₄() ) & ( not F₄() in InnerVertices F₈() implies a4 = s . F₄() ) & ( F₅() in InnerVertices F₈() implies a5 = (Result s') . F₅() ) & ( not F₅() in InnerVertices F₈() implies a5 = s . F₅() ) holds
(Result s) . (Output (1GateCircStr <*F₁(),F₂(),F₃(),F₄(),F₅()*>,F₁₀())) = F₇(a1,a2,a3,a4,a5)
let s' be State of F₉(); :: thesis: ( s' = s | the carrier of F₈() implies for a1, a2, a3, a4, a5 being Element of F₆() st ( F₁() in InnerVertices F₈() implies a1 = (Result s') . F₁() ) & ( not F₁() in InnerVertices F₈() implies a1 = s . F₁() ) & ( F₂() in InnerVertices F₈() implies a2 = (Result s') . F₂() ) & ( not F₂() in InnerVertices F₈() implies a2 = s . F₂() ) & ( F₃() in InnerVertices F₈() implies a3 = (Result s') . F₃() ) & ( not F₃() in InnerVertices F₈() implies a3 = s . F₃() ) & ( F₄() in InnerVertices F₈() implies a4 = (Result s') . F₄() ) & ( not F₄() in InnerVertices F₈() implies a4 = s . F₄() ) & ( F₅() in InnerVertices F₈() implies a5 = (Result s') . F₅() ) & ( not F₅() in InnerVertices F₈() implies a5 = s . F₅() ) holds
(Result s) . (Output (1GateCircStr <*F₁(),F₂(),F₃(),F₄(),F₅()*>,F₁₀())) = F₇(a1,a2,a3,a4,a5) )
assume A3: s' = s | the carrier of F₈() ; :: thesis: for a1, a2, a3, a4, a5 being Element of F₆() st ( F₁() in InnerVertices F₈() implies a1 = (Result s') . F₁() ) & ( not F₁() in InnerVertices F₈() implies a1 = s . F₁() ) & ( F₂() in InnerVertices F₈() implies a2 = (Result s') . F₂() ) & ( not F₂() in InnerVertices F₈() implies a2 = s . F₂() ) & ( F₃() in InnerVertices F₈() implies a3 = (Result s') . F₃() ) & ( not F₃() in InnerVertices F₈() implies a3 = s . F₃() ) & ( F₄() in InnerVertices F₈() implies a4 = (Result s') . F₄() ) & ( not F₄() in InnerVertices F₈() implies a4 = s . F₄() ) & ( F₅() in InnerVertices F₈() implies a5 = (Result s') . F₅() ) & ( not F₅() in InnerVertices F₈() implies a5 = s . F₅() ) holds
(Result s) . (Output (1GateCircStr <*F₁(),F₂(),F₃(),F₄(),F₅()*>,F₁₀())) = F₇(a1,a2,a3,a4,a5)
let a1, a2, a3, a4, a5 be Element of F₆(); :: thesis: ( ( F₁() in InnerVertices F₈() implies a1 = (Result s') . F₁() ) & ( not F₁() in InnerVertices F₈() implies a1 = s . F₁() ) & ( F₂() in InnerVertices F₈() implies a2 = (Result s') . F₂() ) & ( not F₂() in InnerVertices F₈() implies a2 = s . F₂() ) & ( F₃() in InnerVertices F₈() implies a3 = (Result s') . F₃() ) & ( not F₃() in InnerVertices F₈() implies a3 = s . F₃() ) & ( F₄() in InnerVertices F₈() implies a4 = (Result s') . F₄() ) & ( not F₄() in InnerVertices F₈() implies a4 = s . F₄() ) & ( F₅() in InnerVertices F₈() implies a5 = (Result s') . F₅() ) & ( not F₅() in InnerVertices F₈() implies a5 = s . F₅() ) implies (Result s) . (Output (1GateCircStr <*F₁(),F₂(),F₃(),F₄(),F₅()*>,F₁₀())) = F₇(a1,a2,a3,a4,a5) )
assume A4: ( ( F₁() in InnerVertices F₈() implies a1 = (Result s') . F₁() ) & ( not F₁() in InnerVertices F₈() implies a1 = s . F₁() ) & ( F₂() in InnerVertices F₈() implies a2 = (Result s') . F₂() ) & ( not F₂() in InnerVertices F₈() implies a2 = s . F₂() ) & ( F₃() in InnerVertices F₈() implies a3 = (Result s') . F₃() ) & ( not F₃() in InnerVertices F₈() implies a3 = s . F₃() ) & ( F₄() in InnerVertices F₈() implies a4 = (Result s') . F₄() ) & ( not F₄() in InnerVertices F₈() implies a4 = s . F₄() ) & ( F₅() in InnerVertices F₈() implies a5 = (Result s') . F₅() ) & ( not F₅() in InnerVertices F₈() implies a5 = s . F₅() ) ) ; :: thesis: (Result s) . (Output (1GateCircStr <*F₁(),F₂(),F₃(),F₄(),F₅()*>,F₁₀())) = F₇(a1,a2,a3,a4,a5)
set S = 1GateCircStr <*F₁(),F₂(),F₃(),F₄(),F₅()*>,F₁₀();
rng <*F₁(),F₂(),F₃(),F₄(),F₅()*> = {F₁(),F₂(),F₃(),F₄(),F₅()} by Th15;
then A5: ( the carrier of (1GateCircStr <*F₁(),F₂(),F₃(),F₄(),F₅()*>,F₁₀()) = (rng <*F₁(),F₂(),F₃(),F₄(),F₅()*>) \/ {[<*F₁(),F₂(),F₃(),F₄(),F₅()*>,F₁₀()]} & InputVertices (1GateCircStr <*F₁(),F₂(),F₃(),F₄(),F₅()*>,F₁₀()) = rng <*F₁(),F₂(),F₃(),F₄(),F₅()*> & F₁() in rng <*F₁(),F₂(),F₃(),F₄(),F₅()*> & F₂() in rng <*F₁(),F₂(),F₃(),F₄(),F₅()*> & F₃() in rng <*F₁(),F₂(),F₃(),F₄(),F₅()*> & F₄() in rng <*F₁(),F₂(),F₃(),F₄(),F₅()*> & F₅() in rng <*F₁(),F₂(),F₃(),F₄(),F₅()*> ) by CIRCCOMB:49, CIRCCOMB:def 6, ENUMSET1:def 3;
then ( F₁() in the carrier of (1GateCircStr <*F₁(),F₂(),F₃(),F₄(),F₅()*>,F₁₀()) & F₂() in the carrier of (1GateCircStr <*F₁(),F₂(),F₃(),F₄(),F₅()*>,F₁₀()) & F₃() in the carrier of (1GateCircStr <*F₁(),F₂(),F₃(),F₄(),F₅()*>,F₁₀()) & F₄() in the carrier of (1GateCircStr <*F₁(),F₂(),F₃(),F₄(),F₅()*>,F₁₀()) & F₅() in the carrier of (1GateCircStr <*F₁(),F₂(),F₃(),F₄(),F₅()*>,F₁₀()) & the carrier of (F₈() +* (1GateCircStr <*F₁(),F₂(),F₃(),F₄(),F₅()*>,F₁₀())) = the carrier of F₈() \/ the carrier of (1GateCircStr <*F₁(),F₂(),F₃(),F₄(),F₅()*>,F₁₀()) ) by CIRCCOMB:def 2;
then A6: ( F₁() in the carrier of (F₈() +* (1GateCircStr <*F₁(),F₂(),F₃(),F₄(),F₅()*>,F₁₀())) & F₂() in the carrier of (F₈() +* (1GateCircStr <*F₁(),F₂(),F₃(),F₄(),F₅()*>,F₁₀())) & F₃() in the carrier of (F₈() +* (1GateCircStr <*F₁(),F₂(),F₃(),F₄(),F₅()*>,F₁₀())) & F₄() in the carrier of (F₈() +* (1GateCircStr <*F₁(),F₂(),F₃(),F₄(),F₅()*>,F₁₀())) & F₅() in the carrier of (F₈() +* (1GateCircStr <*F₁(),F₂(),F₃(),F₄(),F₅()*>,F₁₀())) ) by XBOOLE_0:def 3;
InnerVertices (1GateCircStr <*F₁(),F₂(),F₃(),F₄(),F₅()*>,F₁₀()) = {(Output (1GateCircStr <*F₁(),F₂(),F₃(),F₄(),F₅()*>,F₁₀()))} by Th17;
then A7: InputVertices F₈() misses InnerVertices (1GateCircStr <*F₁(),F₂(),F₃(),F₄(),F₅()*>,F₁₀()) by A2, ZFMISC_1:56;
F₉() tolerates 1GateCircuit <*F₁(),F₂(),F₃(),F₄(),F₅()*>,F₁₀() by Th30;
then the Sorts of F₉() tolerates the Sorts of (1GateCircuit <*F₁(),F₂(),F₃(),F₄(),F₅()*>,F₁₀()) by CIRCCOMB:def 3;
then reconsider s1 = (Following s,(stabilization-time s')) | the carrier of (1GateCircStr <*F₁(),F₂(),F₃(),F₄(),F₅()*>,F₁₀()) as State of (1GateCircuit <*F₁(),F₂(),F₃(),F₄(),F₅()*>,F₁₀()) by CIRCCOMB:33;
A8: ( s is stabilizing & s' is stabilizing & s1 is stabilizing ) by Def2;
F₈() tolerates 1GateCircStr <*F₁(),F₂(),F₃(),F₄(),F₅()*>,F₁₀() by CIRCCOMB:55;
then A9: InputVertices (F₈() +* (1GateCircStr <*F₁(),F₂(),F₃(),F₄(),F₅()*>,F₁₀())) = (InputVertices F₈()) \/ ((InputVertices (1GateCircStr <*F₁(),F₂(),F₃(),F₄(),F₅()*>,F₁₀())) \ (InnerVertices F₈())) by A7, FACIRC_1:4;
( F₁() in InnerVertices F₈() or F₁() in (InputVertices (1GateCircStr <*F₁(),F₂(),F₃(),F₄(),F₅()*>,F₁₀())) \ (InnerVertices F₈()) ) by A5, XBOOLE_0:def 5;
then A10: ( F₁() in InnerVertices F₈() or F₁() in InputVertices (F₈() +* (1GateCircStr <*F₁(),F₂(),F₃(),F₄(),F₅()*>,F₁₀())) ) by A9, XBOOLE_0:def 3;
( F₂() in InnerVertices F₈() or F₂() in (InputVertices (1GateCircStr <*F₁(),F₂(),F₃(),F₄(),F₅()*>,F₁₀())) \ (InnerVertices F₈()) ) by A5, XBOOLE_0:def 5;
then A11: ( F₂() in InnerVertices F₈() or F₂() in InputVertices (F₈() +* (1GateCircStr <*F₁(),F₂(),F₃(),F₄(),F₅()*>,F₁₀())) ) by A9, XBOOLE_0:def 3;
( F₃() in InnerVertices F₈() or F₃() in (InputVertices (1GateCircStr <*F₁(),F₂(),F₃(),F₄(),F₅()*>,F₁₀())) \ (InnerVertices F₈()) ) by A5, XBOOLE_0:def 5;
then A12: ( F₃() in InnerVertices F₈() or F₃() in InputVertices (F₈() +* (1GateCircStr <*F₁(),F₂(),F₃(),F₄(),F₅()*>,F₁₀())) ) by A9, XBOOLE_0:def 3;
( F₄() in InnerVertices F₈() or F₄() in (InputVertices (1GateCircStr <*F₁(),F₂(),F₃(),F₄(),F₅()*>,F₁₀())) \ (InnerVertices F₈()) ) by A5, XBOOLE_0:def 5;
then A13: ( F₄() in InnerVertices F₈() or F₄() in InputVertices (F₈() +* (1GateCircStr <*F₁(),F₂(),F₃(),F₄(),F₅()*>,F₁₀())) ) by A9, XBOOLE_0:def 3;
( F₅() in InnerVertices F₈() or F₅() in (InputVertices (1GateCircStr <*F₁(),F₂(),F₃(),F₄(),F₅()*>,F₁₀())) \ (InnerVertices F₈()) ) by A5, XBOOLE_0:def 5;
then A14: ( F₅() in InnerVertices F₈() or F₅() in InputVertices (F₈() +* (1GateCircStr <*F₁(),F₂(),F₃(),F₄(),F₅()*>,F₁₀())) ) by A9, XBOOLE_0:def 3;
(Following s,(stabilization-time s')) | the carrier of F₈() = Following s',(stabilization-time s') by A3, A7, Th30, CIRCCMB2:14
.= Result s' by A8, Th2 ;
then A15: ( a1 = (Following s,(stabilization-time s')) . F₁() & a2 = (Following s,(stabilization-time s')) . F₂() & a3 = (Following s,(stabilization-time s')) . F₃() & a4 = (Following s,(stabilization-time s')) . F₄() & a5 = (Following s,(stabilization-time s')) . F₅() ) by A4, A10, A11, A12, A13, A14, Th1, FUNCT_1:72;
dom (Following s,(stabilization-time s')) = the carrier of (F₈() +* (1GateCircStr <*F₁(),F₂(),F₃(),F₄(),F₅()*>,F₁₀())) by CIRCUIT1:4;
then A16: (Following s,(stabilization-time s')) * <*F₁(),F₂(),F₃(),F₄(),F₅()*> = <*a1,a2,a3,a4,a5*> by A6, A15, Th38;
A17: [<*F₁(),F₂(),F₃(),F₄(),F₅()*>,F₁₀()] = Output (1GateCircStr <*F₁(),F₂(),F₃(),F₄(),F₅()*>,F₁₀()) by Th16;
the carrier' of (1GateCircStr <*F₁(),F₂(),F₃(),F₄(),F₅()*>,F₁₀()) = {[<*F₁(),F₂(),F₃(),F₄(),F₅()*>,F₁₀()]} by CIRCCOMB:def 6;
then ( [<*F₁(),F₂(),F₃(),F₄(),F₅()*>,F₁₀()] in {[<*F₁(),F₂(),F₃(),F₄(),F₅()*>,F₁₀()]} & the carrier' of (F₈() +* (1GateCircStr <*F₁(),F₂(),F₃(),F₄(),F₅()*>,F₁₀())) = the carrier' of F₈() \/ {[<*F₁(),F₂(),F₃(),F₄(),F₅()*>,F₁₀()]} ) by CIRCCOMB:def 2, TARSKI:def 1;
then reconsider g = [<*F₁(),F₂(),F₃(),F₄(),F₅()*>,F₁₀()] as Gate of (F₈() +* (1GateCircStr <*F₁(),F₂(),F₃(),F₄(),F₅()*>,F₁₀())) by XBOOLE_0:def 3;
A18: the_result_sort_of g = the ResultSort of (F₈() +* (1GateCircStr <*F₁(),F₂(),F₃(),F₄(),F₅()*>,F₁₀())) . g by MSUALG_1:def 7
.= g by CIRCCOMB:52 ;
g = [(the Arity of (F₈() +* (1GateCircStr <*F₁(),F₂(),F₃(),F₄(),F₅()*>,F₁₀())) . g),(g `2 )] by CIRCCOMB:def 8;
then A19: <*F<sub>1</sub>(),F<sub>2</sub>(),F<sub>3</sub>(),F<sub>4</sub>(),F<sub>5</sub>()*> = the Arity of (F<sub>8</sub>() +* (1GateCircStr <*F<sub>1</sub>(),F<sub>2</sub>(),F<sub>3</sub>(),F<sub>4</sub>(),F<sub>5</sub>()*>,F<sub>10</sub>())) . g by ZFMISC_1:33
.= the_arity_of g by MSUALG_1:def 6 ;
A20: g `2 = F₁₀() by MCART_1:7;
stabilization-time s <= 1 + (stabilization-time s') by A2, A3, Th52;
hence (Result s) . (Output (1GateCircStr <*F₁(),F₂(),F₃(),F₄(),F₅()*>,F₁₀())) = (Following s,(1 + (stabilization-time s'))) . (Output (1GateCircStr <*F₁(),F₂(),F₃(),F₄(),F₅()*>,F₁₀())) by A8, Th5
.= (Following (Following s,(stabilization-time s'))) . g by A17, FACIRC_1:12
.= F₁₀() . ((Following s,(stabilization-time s')) * <*F₁(),F₂(),F₃(),F₄(),F₅()*>) by A18, A19, A20, FACIRC_1:34
.= F₇(a1,a2,a3,a4,a5) by A1, A16 ;
:: thesis: verum