reserve IIG for Circuit-like non void non empty ManySortedSign;
reserve IIG for monotonic Circuit-like non void non empty ManySortedSign;

theorem
  for IIG being finite monotonic Circuit-like non void non empty
ManySortedSign, A being non-empty Circuit of IIG, v being Vertex of IIG holds
  depth(v,A) <= depth A
proof
  let IIG be finite monotonic Circuit-like non void non empty ManySortedSign
  , A be non-empty Circuit of IIG, v be Vertex of IIG;
  consider Ds being finite non empty Subset of NAT such that
A1: Ds = { depth(v9,A) where v9 is Element of IIG : v9 in the carrier of
  IIG } and
A2: depth A = max Ds by Def7;
  reconsider Y = Ds as finite non empty real-membered set;
  depth(v,A) in Y by A1;
  hence thesis by A2,XXREAL_2:def 8;
end;
