reserve A for preIfWhileAlgebra,
  C,I,J for Element of A;
reserve S for non empty set,
  T for Subset of S,
  s for Element of S;

theorem
  for X being disjoint_with_NAT non empty set
  for p being FinSequence of FreeUnivAlgNSG(ECIW-signature,X)
  for x being Element of X
  st x-tree p is Element of FreeUnivAlgNSG(ECIW-signature,X) holds p = {}
proof
  let X be disjoint_with_NAT non empty set;
  set S = ECIW-signature;
  set A = FreeUnivAlgNSG(S, X);
  let p be FinSequence of FreeUnivAlgNSG(ECIW-signature,X);
  let x be Element of X;
  assume x-tree p is Element of FreeUnivAlgNSG(ECIW-signature,X);
  then reconsider I = x-tree p as Element of A;
  now
    given n being Nat, p being FinSequence of A such that
A1: n in Seg 4 and
A2: I = n-tree p and len p = ECIW-signature.n;
A3: x = n by A2,TREES_4:15;
    X misses NAT by FREEALG:def 1;
    hence contradiction by A1,A3,XBOOLE_0:3;
  end;
  then consider y being Element of X such that
A4: I = root-tree y by Th56;
  x-tree p = y-tree {} by A4,TREES_4:20;
  hence thesis by TREES_4:15;
end;
