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 Th61:
  for X being disjoint_with_NAT non empty set
  for I1,I2 being Element of FreeUnivAlgNSG(ECIW-signature,X)
  holds I1\;I2 <> I1 & I1\;I2 <> I2
proof
  let X be disjoint_with_NAT non empty set;
  set S = ECIW-signature;
  set G = DTConUA(S,X);
  set A = FreeUnivAlgNSG(S,X);
  let I1,I2 be Element of A;
  set p = <*I1,I2*>;
  rng p c= FinTrees the carrier of G by XBOOLE_1:1;
  then
A1: p is FinSequence of FinTrees the carrier of G by FINSEQ_1:def 4;
A2: rng p = {I1,I2} by FINSEQ_2:127;
  then
A3: I1 in rng p by TARSKI:def 2;
A4: I2 in rng p by A2,TARSKI:def 2;
  I1\;I2 = 2-tree(I1,I2) by Th59
    .= 2-tree<*I1,I2*>;
  hence thesis by A1,A3,A4,Th3;
end;
