let IIG be non empty finite non void Circuit-like monotonic ManySortedSign ; :: thesis: for SCS being non-empty Circuit of non-empty
for InpFs being InputFuncs of st commute InpFs is constant & not InputVertices IIG is empty holds
for s1, s2 being State of holds (Computation s1,InpFs) . (depth SCS) = (Computation s2,InpFs) . (depth SCS)
let SCS be non-empty Circuit of non-empty ; :: thesis: for InpFs being InputFuncs of st commute InpFs is constant & not InputVertices IIG is empty holds
for s1, s2 being State of holds (Computation s1,InpFs) . (depth SCS) = (Computation s2,InpFs) . (depth SCS)
let InpFs be InputFuncs of ; :: thesis: ( commute InpFs is constant & not InputVertices IIG is empty implies for s1, s2 being State of 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 holds (Computation s1,InpFs) . (depth SCS) = (Computation s2,InpFs) . (depth SCS)
IIG is with_input_V by A2, MSAFREE2:def 4;
then reconsider iv = (commute InpFs) . 0 as InputValues of SCS by CIRCUIT1:2;
reconsider dSCS = depth SCS as Element of NAT by ORDINAL1:def 13;
let s1, s2 be State of ; :: thesis: (Computation s1,InpFs) . (depth SCS) = (Computation s2,InpFs) . (depth SCS)
reconsider Cs1 = (Computation s1,InpFs) . dSCS as State of ;
reconsider Cs2 = (Computation s2,InpFs) . dSCS as State of ;

now

thus the carrier of IIG = dom Cs1 by CIRCUIT1:4; :: thesis: ( the carrier of IIG = dom Cs2 & ( for x being set st x in the carrier of IIG holds
Cs1 . x = Cs2 . x ) )
thus the carrier of IIG = dom Cs2 by CIRCUIT1:4; :: thesis: for x being set st x in the carrier of IIG holds
Cs1 . x = Cs2 . x
let x be set ; :: 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 x' = x as Vertex of ;
Cs1 . x' = IGValue x',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) by FUNCT_1:9; :: thesis: verum