let IIG be non empty finite non void Circuit-like monotonic ManySortedSign ; :: thesis: for SCS being non-empty Circuit of IIG
for InpFs being InputFuncs of SCS st commute InpFs is constant & not InputVertices IIG is empty holds
for s1, s2 being State of SCS holds (Computation (s1,InpFs)) . (depth SCS) = (Computation (s2,InpFs)) . (depth SCS)
let SCS be non-empty Circuit of IIG; :: thesis: for InpFs being InputFuncs of SCS st commute InpFs is constant & not InputVertices IIG is empty holds
for s1, s2 being State of SCS holds (Computation (s1,InpFs)) . (depth SCS) = (Computation (s2,InpFs)) . (depth SCS)
let InpFs be InputFuncs of SCS; :: thesis: ( commute InpFs is constant & not InputVertices IIG is empty implies for s1, s2 being State of SCS holds (Computation (s1,InpFs)) . (depth SCS) = (Computation (s2,InpFs)) . (depth SCS) )
assume that
A1: commute InpFs is constant and
A2: not InputVertices IIG is empty ; :: thesis: for s1, s2 being State of SCS holds (Computation (s1,InpFs)) . (depth SCS) = (Computation (s2,InpFs)) . (depth SCS)
IIG is with_input_V by A2;
then reconsider iv = (commute InpFs) . 0 as InputValues of SCS by CIRCUIT1:2;
reconsider dSCS = depth SCS as Element of NAT by ORDINAL1:def 12;
let s1, s2 be State of SCS; :: thesis: (Computation (s1,InpFs)) . (depth SCS) = (Computation (s2,InpFs)) . (depth SCS)
reconsider Cs1 = (Computation (s1,InpFs)) . dSCS as State of SCS ;
reconsider Cs2 = (Computation (s2,InpFs)) . dSCS as State of SCS ;

now :: thesis: ( the carrier of IIG = dom Cs1 & the carrier of IIG = dom Cs2 & ( for x being object st x in the carrier of IIG holds
Cs1 . x = Cs2 . x ) )

thus the carrier of IIG = dom Cs1 by CIRCUIT1:3; :: thesis: ( the carrier of IIG = dom Cs2 & ( for x being object st x in the carrier of IIG holds
Cs1 . x = Cs2 . x ) )
thus the carrier of IIG = dom Cs2 by CIRCUIT1:3; :: thesis: for x being object st x in the carrier of IIG holds
Cs1 . x = Cs2 . x
let x be object ; :: thesis: ( x in the carrier of IIG implies Cs1 . x = Cs2 . x )
assume x in the carrier of IIG ; :: thesis: Cs1 . x = Cs2 . x
then reconsider x9 = x as Vertex of IIG ;
Cs1 . x9 = IGValue (x9,iv) by A1, A2, Th17;
hence Cs1 . x = Cs2 . x by A1, A2, Th17; :: thesis: verum

end;

hence (Computation (s1,InpFs)) . (depth SCS) = (Computation (s2,InpFs)) . (depth SCS) ; :: thesis: verum