let s be State of (F₇() +* (1GateCircuit <*F₁(),F₂(),F₃()*>,F₈())); :: thesis: for s' being State of F₇() st s' = s | the carrier of F₆() holds
for a1, a2, a3 being Element of F₄() st ( F₁() in InnerVertices F₆() implies a1 = (Result s') . F₁() ) & ( not F₁() in InnerVertices F₆() implies a1 = s . F₁() ) & ( F₂() in InnerVertices F₆() implies a2 = (Result s') . F₂() ) & ( not F₂() in InnerVertices F₆() implies a2 = s . F₂() ) & ( F₃() in InnerVertices F₆() implies a3 = (Result s') . F₃() ) & ( not F₃() in InnerVertices F₆() implies a3 = s . F₃() ) holds
(Result s) . (Output (1GateCircStr <*F₁(),F₂(),F₃()*>,F₈())) = F₅(a1,a2,a3)
let s' be State of F₇(); :: thesis: ( s' = s | the carrier of F₆() implies for a1, a2, a3 being Element of F₄() st ( F₁() in InnerVertices F₆() implies a1 = (Result s') . F₁() ) & ( not F₁() in InnerVertices F₆() implies a1 = s . F₁() ) & ( F₂() in InnerVertices F₆() implies a2 = (Result s') . F₂() ) & ( not F₂() in InnerVertices F₆() implies a2 = s . F₂() ) & ( F₃() in InnerVertices F₆() implies a3 = (Result s') . F₃() ) & ( not F₃() in InnerVertices F₆() implies a3 = s . F₃() ) holds
(Result s) . (Output (1GateCircStr <*F₁(),F₂(),F₃()*>,F₈())) = F₅(a1,a2,a3) )
assume A3: s' = s | the carrier of F₆() ; :: thesis: for a1, a2, a3 being Element of F₄() st ( F₁() in InnerVertices F₆() implies a1 = (Result s') . F₁() ) & ( not F₁() in InnerVertices F₆() implies a1 = s . F₁() ) & ( F₂() in InnerVertices F₆() implies a2 = (Result s') . F₂() ) & ( not F₂() in InnerVertices F₆() implies a2 = s . F₂() ) & ( F₃() in InnerVertices F₆() implies a3 = (Result s') . F₃() ) & ( not F₃() in InnerVertices F₆() implies a3 = s . F₃() ) holds
(Result s) . (Output (1GateCircStr <*F₁(),F₂(),F₃()*>,F₈())) = F₅(a1,a2,a3)
let a1, a2, a3 be Element of F₄(); :: thesis: ( ( F₁() in InnerVertices F₆() implies a1 = (Result s') . F₁() ) & ( not F₁() in InnerVertices F₆() implies a1 = s . F₁() ) & ( F₂() in InnerVertices F₆() implies a2 = (Result s') . F₂() ) & ( not F₂() in InnerVertices F₆() implies a2 = s . F₂() ) & ( F₃() in InnerVertices F₆() implies a3 = (Result s') . F₃() ) & ( not F₃() in InnerVertices F₆() implies a3 = s . F₃() ) implies (Result s) . (Output (1GateCircStr <*F₁(),F₂(),F₃()*>,F₈())) = F₅(a1,a2,a3) )
assume A4: ( ( F₁() in InnerVertices F₆() implies a1 = (Result s') . F₁() ) & ( not F₁() in InnerVertices F₆() implies a1 = s . F₁() ) & ( F₂() in InnerVertices F₆() implies a2 = (Result s') . F₂() ) & ( not F₂() in InnerVertices F₆() implies a2 = s . F₂() ) & ( F₃() in InnerVertices F₆() implies a3 = (Result s') . F₃() ) & ( not F₃() in InnerVertices F₆() implies a3 = s . F₃() ) ) ; :: thesis: (Result s) . (Output (1GateCircStr <*F₁(),F₂(),F₃()*>,F₈())) = F₅(a1,a2,a3)
set S = 1GateCircStr <*F₁(),F₂(),F₃()*>,F₈();
rng <*F₁(),F₂(),F₃()*> = {F₁(),F₂(),F₃()} by FINSEQ_2:148;
then A5: ( the carrier of (1GateCircStr <*F₁(),F₂(),F₃()*>,F₈()) = (rng <*F₁(),F₂(),F₃()*>) \/ {[<*F₁(),F₂(),F₃()*>,F₈()]} & InputVertices (1GateCircStr <*F₁(),F₂(),F₃()*>,F₈()) = rng <*F₁(),F₂(),F₃()*> & F₃() in rng <*F₁(),F₂(),F₃()*> & F₂() in rng <*F₁(),F₂(),F₃()*> & F₁() in rng <*F₁(),F₂(),F₃()*> ) by CIRCCOMB:49, CIRCCOMB:def 6, ENUMSET1:def 1;
then ( F₃() in the carrier of (1GateCircStr <*F₁(),F₂(),F₃()*>,F₈()) & F₂() in the carrier of (1GateCircStr <*F₁(),F₂(),F₃()*>,F₈()) & F₁() in the carrier of (1GateCircStr <*F₁(),F₂(),F₃()*>,F₈()) & the carrier of (F₆() +* (1GateCircStr <*F₁(),F₂(),F₃()*>,F₈())) = the carrier of F₆() \/ the carrier of (1GateCircStr <*F₁(),F₂(),F₃()*>,F₈()) ) by CIRCCOMB:def 2;
then A6: ( F₁() in the carrier of (F₆() +* (1GateCircStr <*F₁(),F₂(),F₃()*>,F₈())) & F₂() in the carrier of (F₆() +* (1GateCircStr <*F₁(),F₂(),F₃()*>,F₈())) & F₃() in the carrier of (F₆() +* (1GateCircStr <*F₁(),F₂(),F₃()*>,F₈())) ) by XBOOLE_0:def 3;
InnerVertices (1GateCircStr <*F₁(),F₂(),F₃()*>,F₈()) = {(Output (1GateCircStr <*F₁(),F₂(),F₃()*>,F₈()))} by Th17;
then A7: InputVertices F₆() misses InnerVertices (1GateCircStr <*F₁(),F₂(),F₃()*>,F₈()) by A2, ZFMISC_1:56;
F₇() tolerates 1GateCircuit <*F₁(),F₂(),F₃()*>,F₈() by Th30;
then the Sorts of F₇() tolerates the Sorts of (1GateCircuit <*F₁(),F₂(),F₃()*>,F₈()) by CIRCCOMB:def 3;
then reconsider s1 = (Following s,(stabilization-time s')) | the carrier of (1GateCircStr <*F₁(),F₂(),F₃()*>,F₈()) as State of (1GateCircuit <*F₁(),F₂(),F₃()*>,F₈()) by CIRCCOMB:33;
A8: ( s is stabilizing & s' is stabilizing & s1 is stabilizing ) by Def2;
F₆() tolerates 1GateCircStr <*F₁(),F₂(),F₃()*>,F₈() by CIRCCOMB:55;
then A9: InputVertices (F₆() +* (1GateCircStr <*F₁(),F₂(),F₃()*>,F₈())) = (InputVertices F₆()) \/ ((InputVertices (1GateCircStr <*F₁(),F₂(),F₃()*>,F₈())) \ (InnerVertices F₆())) by A7, FACIRC_1:4;
( F₁() in InnerVertices F₆() or F₁() in (InputVertices (1GateCircStr <*F₁(),F₂(),F₃()*>,F₈())) \ (InnerVertices F₆()) ) by A5, XBOOLE_0:def 5;
then A10: ( F₁() in InnerVertices F₆() or F₁() in InputVertices (F₆() +* (1GateCircStr <*F₁(),F₂(),F₃()*>,F₈())) ) by A9, XBOOLE_0:def 3;
( F₂() in InnerVertices F₆() or F₂() in (InputVertices (1GateCircStr <*F₁(),F₂(),F₃()*>,F₈())) \ (InnerVertices F₆()) ) by A5, XBOOLE_0:def 5;
then A11: ( F₂() in InnerVertices F₆() or F₂() in InputVertices (F₆() +* (1GateCircStr <*F₁(),F₂(),F₃()*>,F₈())) ) by A9, XBOOLE_0:def 3;
( F₃() in InnerVertices F₆() or F₃() in (InputVertices (1GateCircStr <*F₁(),F₂(),F₃()*>,F₈())) \ (InnerVertices F₆()) ) by A5, XBOOLE_0:def 5;
then A12: ( F₃() in InnerVertices F₆() or F₃() in InputVertices (F₆() +* (1GateCircStr <*F₁(),F₂(),F₃()*>,F₈())) ) by A9, XBOOLE_0:def 3;
(Following s,(stabilization-time s')) | the carrier of F₆() = Following s',(stabilization-time s') by A3, A7, Th30, CIRCCMB2:14
.= Result s' by A8, Th2 ;
then A13: ( a1 = (Following s,(stabilization-time s')) . F₁() & a2 = (Following s,(stabilization-time s')) . F₂() & a3 = (Following s,(stabilization-time s')) . F₃() ) by A4, A10, A11, A12, Th1, FUNCT_1:72;
dom (Following s,(stabilization-time s')) = the carrier of (F₆() +* (1GateCircStr <*F₁(),F₂(),F₃()*>,F₈())) by CIRCUIT1:4;
then A14: (Following s,(stabilization-time s')) * <*F₁(),F₂(),F₃()*> = <*a1,a2,a3*> by A6, A13, FINSEQ_2:146;
A15: [<*F₁(),F₂(),F₃()*>,F₈()] = Output (1GateCircStr <*F₁(),F₂(),F₃()*>,F₈()) by Th16;
the carrier' of (1GateCircStr <*F₁(),F₂(),F₃()*>,F₈()) = {[<*F₁(),F₂(),F₃()*>,F₈()]} by CIRCCOMB:def 6;
then ( [<*F₁(),F₂(),F₃()*>,F₈()] in {[<*F₁(),F₂(),F₃()*>,F₈()]} & the carrier' of (F₆() +* (1GateCircStr <*F₁(),F₂(),F₃()*>,F₈())) = the carrier' of F₆() \/ {[<*F₁(),F₂(),F₃()*>,F₈()]} ) by CIRCCOMB:def 2, TARSKI:def 1;
then reconsider g = [<*F₁(),F₂(),F₃()*>,F₈()] as Gate of (F₆() +* (1GateCircStr <*F₁(),F₂(),F₃()*>,F₈())) by XBOOLE_0:def 3;
A16: the_result_sort_of g = the ResultSort of (F₆() +* (1GateCircStr <*F₁(),F₂(),F₃()*>,F₈())) . g by MSUALG_1:def 7
.= g by CIRCCOMB:52 ;
g = [(the Arity of (F₆() +* (1GateCircStr <*F₁(),F₂(),F₃()*>,F₈())) . g),(g `2 )] by CIRCCOMB:def 8;
then A17: <*F<sub>1</sub>(),F<sub>2</sub>(),F<sub>3</sub>()*> = the Arity of (F<sub>6</sub>() +* (1GateCircStr <*F<sub>1</sub>(),F<sub>2</sub>(),F<sub>3</sub>()*>,F<sub>8</sub>())) . g by ZFMISC_1:33
.= the_arity_of g by MSUALG_1:def 6 ;
A18: g `2 = F₈() by MCART_1:7;
stabilization-time s <= 1 + (stabilization-time s') by A2, A3, Th52;
hence (Result s) . (Output (1GateCircStr <*F₁(),F₂(),F₃()*>,F₈())) = (Following s,(1 + (stabilization-time s'))) . (Output (1GateCircStr <*F₁(),F₂(),F₃()*>,F₈())) by A8, Th5
.= (Following (Following s,(stabilization-time s'))) . g by A15, FACIRC_1:12
.= F₈() . ((Following s,(stabilization-time s')) * <*F₁(),F₂(),F₃()*>) by A16, A17, A18, FACIRC_1:34
.= F₅(a1,a2,a3) by A1, A14 ;
:: thesis: verum