
theorem
  for S being non void non empty ManySortedSign, A being non-empty
MSAlgebra over S, v being Vertex of S, e being Element of (the Sorts of FreeEnv
  A).v st v in InputVertices S ex x being Element of (the Sorts of A).v st e =
  root-tree [x, v]
proof
  let S be non void non empty ManySortedSign, A be non-empty MSAlgebra over S,
  v be Vertex of S, e be Element of (the Sorts of FreeEnv A).v;
  FreeEnv A = MSAlgebra (# FreeSort(the Sorts of A), FreeOper(the Sorts of
    A) #) by MSAFREE:def 14;
  then e in (FreeSort(the Sorts of A)).v;
  then e in FreeSort(the Sorts of A, v) by MSAFREE:def 11;
  then
  e 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 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(the Sorts of A)) such that
A1: a = e and
A2: (ex x being set st x in (the Sorts of A).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;
  assume v in InputVertices S;
  then consider x being set such that
A3: x in (the Sorts of A).v and
A4: a = root-tree [x,v] by A2,Th2;
  reconsider x as Element of (the Sorts of A).v by A3;
  take x;
  thus thesis by A1,A4;
end;
