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

theorem Th7:
  for IIG for A being finite-yielding non-empty MSAlgebra over IIG,
  v being Element of IIG, e being Element of (the Sorts of FreeEnv A).v st 1 <
  card e ex o being OperSymbol of IIG st e.{} = [o,the carrier of IIG]
proof
  let IIG;
  let A be finite-yielding non-empty MSAlgebra over IIG, v be Element of IIG,
  e be Element of (the Sorts of FreeEnv A).v;
  set X = the Sorts of A;
  assume
A1: 1 < card e;
A2: (FreeSort X).v = FreeSort(X, v) & FreeSort (X, v) = {a where a is
  Element of TS(DTConMSA(X)): (ex x being set st x in X.v & a = root-tree[x,v])
  or ex o being OperSymbol of IIG st [o,the carrier of IIG] = a.{} &
  the_result_sort_of o = v} by MSAFREE:def 10,def 11;
  e in (the Sorts of FreeMSA X).v & FreeMSA X = MSAlgebra (# FreeSort(X),
    FreeOper(X) #) by MSAFREE:def 14;
  then
  ex a being Element of TS(DTConMSA(X)) st a = e &( (ex x being set st x in
X.v & a = root-tree[x,v]) or ex o being OperSymbol of IIG st [o,the carrier of
  IIG] = a.{} & the_result_sort_of o = v) by A2;
  hence thesis by A1,PRE_CIRC:19;
end;
