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

theorem
  for IIG for A being finite-yielding non-empty MSAlgebra over IIG, v,
  v1 being SortSymbol of IIG st v in InnerVertices IIG & v1 in rng the_arity_of
  action_at v holds depth(v1,A) < depth(v,A)
proof
  let IIG;
  let A be finite-yielding non-empty MSAlgebra over IIG, v, v1 be SortSymbol
  of IIG;
  assume that
A1: v in InnerVertices IIG and
A2: v1 in rng the_arity_of action_at v;
  size(v1,A) > 0 by Th15;
  then
A3: 0 + 1 <= size(v1,A) by NAT_1:13;
  size(v1,A) < size(v,A) by A1,A2,Th14;
  then
A4: not v in InputVertices IIG \/ SortsWithConstants IIG by A3,Th10;
  then
A5: not v in SortsWithConstants IIG by XBOOLE_0:def 3;
  then
A6: v in (InnerVertices IIG \ SortsWithConstants IIG) by A1,XBOOLE_0:def 5;
  consider s1 being finite non empty Subset of NAT such that
A7: s1 = the set of all
 depth t1 where t1 is Element of (the Sorts of FreeEnv A).v1  and
A8: depth(v1,A) = max s1 by Def6;
  reconsider Y1 = s1 as finite non empty real-membered set;
  max Y1 in the set of all
 depth t1 where t1 is Element of (the Sorts of FreeEnv A ) .
v1 by A7,XXREAL_2:def 8;
  then consider t1 being Element of (the Sorts of FreeEnv A).v1 such that
A9: depth t1 = max Y1;
  reconsider av = action_at v as OperSymbol of IIG;
  consider s being finite non empty Subset of NAT such that
A10: s = the set of all
 depth t where t is Element of (the Sorts of FreeEnv A).v  and
A11: depth(v,A) = max s by Def6;
  consider x being object such that
A12: x in dom the_arity_of av and
A13: v1 = (the_arity_of av).x by A2,FUNCT_1:def 3;
  reconsider Y = s as finite non empty real-membered set;
  max Y in the set of all
 depth t where t is Element of (the Sorts of FreeEnv A).v
 by A10,XXREAL_2:def 8;
  then consider t being Element of (the Sorts of FreeEnv A).v such that
A14: depth t = max Y;
  FreeEnv A = MSAlgebra (# FreeSort the Sorts of A, FreeOper the Sorts of
    A #) by MSAFREE:def 14;
  then (the Sorts of FreeEnv A).v = FreeSort(the Sorts of A,v) by
MSAFREE:def 11;
  then t in FreeSort(the Sorts of A,v);
  then
A15: t in {a where a is Element of TS(DTConMSA(the Sorts of A)): (ex x being
  set st x in (the Sorts of A).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;
  reconsider k = x as Element of NAT by A12;
  reconsider k1 = k - 1 as Element of NAT by A12,FINSEQ_3:26;
  reconsider f = <*k1*> as FinSequence of NAT;
A16: k1 + 1 = k;
  reconsider tft = t with-replacement (f,t1) as DecoratedTree;
  consider e9 being Element of (the Sorts of FreeMSA the Sorts of A).v1 such
  that
A17: t1 = e9 and
A18: depth t1 = depth e9 by Def5;
  consider dt19 being finite DecoratedTree, t19 being finite Tree such that
A19: dt19 = e9 & t19 = dom dt19 and
A20: depth e9 = height t19 by MSAFREE2:def 14;
  consider a being Element of TS(DTConMSA(the Sorts of A)) such that
A21: a = t and
A22: (ex x being set st x in (the Sorts of A).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 A15;
A23: not v in InputVertices IIG by A4,XBOOLE_0:def 3;
  now
    given x being set such that
    x in (the Sorts of A).v and
A24: a = root-tree[x,v];
    consider e9 being Element of (the Sorts of FreeMSA the Sorts of A).v such
    that
A25: t = e9 & depth t = depth e9 by Def5;
    ex dta being finite DecoratedTree, ta being finite Tree st dta = e9 &
    ta = dom dta & depth e9 = height ta by MSAFREE2:def 14;
    then depth t = 0 by A21,A24,A25,TREES_1:42,TREES_4:3;
    hence contradiction by A11,A14,A23,A5,Th18;
  end;
  then consider o being OperSymbol of IIG such that
A26: [o,the carrier of IIG] = a.{} and
A27: the_result_sort_of o = v by A22;
  NonTerminals(DTConMSA(the Sorts of A)) = [:the carrier' of IIG,{the
  carrier of IIG}:] & the carrier of IIG in {the carrier of IIG} by MSAFREE:6
,TARSKI:def 1;
  then reconsider
  o9 = [o,the carrier of IIG] as NonTerminal of DTConMSA(the Sorts
  of A) by ZFMISC_1:87;
  consider q being FinSequence of TS(DTConMSA(the Sorts of A)) such that
A28: a = o9-tree q and
  o9 ==> roots q by A26,DTCONSTR:10;
  consider q9 being DTree-yielding FinSequence such that
A29: q = q9 and
A30: dom a = tree(doms q9) by A28,TREES_4:def 4;
A31: t = [av,the carrier of IIG]-tree q9 by A1,A21,A27,A28,A29,MSAFREE2:def 7;
A32: o = av by A1,A27,MSAFREE2:def 7;
  then
A33: len q9 = len the_arity_of av by A21,A27,A28,A29,MSAFREE2:10;
  then
A34: k in dom q9 by A12,FINSEQ_3:29;
A35: dom q9 = dom the_arity_of av by A33,FINSEQ_3:29;
  then consider qq being DTree-yielding FinSequence such that
A36: t with-replacement (f,t1) = o9-tree qq and
A37: len qq = len q9 and
  qq.(k1+1) = t1 and
  for i being Nat st i in dom q9 & i <> k1+1 holds qq.i = q9.i
  by A21,A28,A29,A30,A12,PRE_CIRC:13,15;
A38: k in dom qq by A12,A33,A37,FINSEQ_3:29;
  q9.k in (the Sorts of FreeEnv A).v1 by A21,A27,A28,A29,A12,A13,A32,
MSAFREE2:11;
  then reconsider tft as Element of (the Sorts of FreeEnv A).v by A6,A34,A16
,A31,Th6;
  reconsider dtft = depth tft as Real;
  dtft in Y by A10;
  then
A39: dtft <= depth t by A14,XXREAL_2:def 8;
  consider e9 being Element of (the Sorts of FreeMSA the Sorts of A).v such
  that
A40: tft = e9 and
A41: depth tft = depth e9 by Def5;
  consider dttft being finite DecoratedTree, ttft being finite Tree such that
A42: dttft = e9 & ttft = dom dttft and
A43: depth e9 = height ttft by MSAFREE2:def 14;
  ex qq9 being DTree-yielding FinSequence st qq = qq9 & dom tft = tree(
  doms qq9) by A36,TREES_4:def 4;
  then reconsider f9 = f as Element of ttft by A16,A40,A42,A38,PRE_CIRC:13;
  <*k1*> in tree(doms q9) by A12,A35,A16,PRE_CIRC:13;
  then dom tft = dom t with-replacement (f,dom t1) by A21,A30,TREES_2:def 11;
  then f9 <> {} & ttft|f = t19 by A17,A19,A21,A30,A34,A16,A40,A42,PRE_CIRC:13
,TREES_1:33;
  hence thesis by A11,A14,A8,A9,A18,A20,A39,A41,A43,TREES_1:48,XXREAL_0:2;
end;
