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

theorem Th15:
  for IIG for A being finite-yielding non-empty MSAlgebra over IIG
  , v being SortSymbol of IIG holds size(v,A) > 0
proof
  let IIG;
  let A be finite-yielding non-empty MSAlgebra over IIG, v be SortSymbol of
  IIG;
  consider s being finite non empty Subset of NAT such that
A1: s = the set of all
 card t where t is Element of (the Sorts of FreeEnv A).v  and
A2: size(v,A) = max s by Def4;
  reconsider Y = s as finite non empty real-membered set;
  max Y in the set of all
 card t where t is Element of (the Sorts of FreeEnv A).v  by A1,XXREAL_2:def 8;
  then ex t being Element of (the Sorts of FreeEnv A).v st card t = max Y;
  hence thesis by A2;
end;
