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 I1,I2,J1,J2 being Element of FreeUnivAlgNSG(ECIW-signature,X)
  holds I1\;I2 = J1\;J2 implies I1 = J1 & I2 = J2
proof
  let X be disjoint_with_NAT non empty set;
  set S = ECIW-signature;
  set A = FreeUnivAlgNSG(S,X);
  let I1,I2,J1,J2 be Element of A;
A1: I1\;I2 = 2-tree(I1,I2) by Th59;
  J1\;J2 = 2-tree(J1,J2) by Th59;
  then I1\;I2 = J1\;J2 implies <*I1,I2*> = <*J1,J2*> by A1,TREES_4:15;
  hence thesis by FINSEQ_1:77;
end;
