
theorem
  for S being non void non empty ManySortedSign, X being non-empty
ManySortedSet of the carrier of S, v being Vertex of S, o being OperSymbol of S
, e being Element of (the Sorts of FreeMSA X).v st e.{} = [o,the carrier of S]
  ex p being DTree-yielding FinSequence st len p = len the_arity_of o & for i
being Nat st i in dom p holds p.i in (the Sorts of FreeMSA X).((the_arity_of o)
  .i)
proof
  let S be non void non empty ManySortedSign, X be non-empty ManySortedSet of
the carrier of S, v be Vertex of S, o be OperSymbol of S, e be Element of (the
  Sorts of FreeMSA X).v such that
A1: e.{} = [o,the carrier of S];
  the carrier of S in {the carrier of S} by TARSKI:def 1;
  then [o,the carrier of S] in [:the carrier' of S,{the carrier of S}:] by
ZFMISC_1:87;
  then reconsider nt = [o,the carrier of S] as NonTerminal of DTConMSA(X) by
MSAFREE:6;
  FreeMSA X = MSAlgebra (# FreeSort X, FreeOper X #) by MSAFREE:def 14;
  then (the Sorts of FreeMSA X).v = FreeSort(X,v) by MSAFREE:def 11;
  then e in FreeSort(X,v);
  then
  e in {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 S st [o,the carrier of S]
  = a.{} & the_result_sort_of o = v} by MSAFREE:def 10;
  then consider a being Element of TS(DTConMSA(X)) such that
A2: a = e and
A3: (ex x being set st x in X.v & a = root-tree [x,v]) or ex o being
  OperSymbol of S st [o,the carrier of S] = a.{} & the_result_sort_of o = v;
  consider x being set such that
A4: x in X.v & a = root-tree[x,v] or ex o being OperSymbol of S st [o,
  the carrier of S] = a.{} & the_result_sort_of o = v by A3;
  consider p being FinSequence of TS(DTConMSA(X)) such that
A5: a = nt-tree p and
  nt ==> roots p by A1,A2,DTCONSTR:10;
  now
    assume a = root-tree[x,v];
    then
A6: a.{} = [x,v] by TREES_4:3;
A7: for X be set holds not X in X;
    a.{} = [o,the carrier of S] by A5,TREES_4:def 4;
    then the carrier of S = v by A6,XTUPLE_0:1;
    hence contradiction by A7;
  end;
  then consider o9 being OperSymbol of S such that
A8: [o9,the carrier of S] = a.{} and
A9: the_result_sort_of o9 = v by A4;
A10: o = o9 by A1,A2,A8,XTUPLE_0:1;
  then
A11: len p = len the_arity_of o by A2,A5,A9,Th10;
  then dom p = Seg len the_arity_of o by FINSEQ_1:def 3
    .= dom the_arity_of o by FINSEQ_1:def 3;
  then for i being Nat st i in dom p holds p.i in (the Sorts of FreeMSA X).((
  the_arity_of o).i) by A2,A5,A9,A10,Th11;
  hence thesis by A11;
end;
