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 C1,C2,I1,I2,J1,J2 being Element of FreeUnivAlgNSG(ECIW-signature,X)
  st if-then-else(C1,I1,I2) = if-then-else(C2,J1,J2)
  holds C1 = C2 & I1 = J1 & I2 = J2
proof
  let X be disjoint_with_NAT non empty set;
  set S = ECIW-signature;
  set A = FreeUnivAlgNSG(S,X);
  let C1,C2,I1,I2,J1,J2 be Element of A;
A1: if-then-else(C1,I1,I2) = 3-tree<*C1,I1,I2*> by Th63;
  if-then-else(C2,J1,J2) = 3-tree<*C2,J1,J2*> by Th63;
  then if-then-else(C1,I1,I2) = if-then-else(C2,J1,J2)
  implies <*C1,I1,I2*> = <*C2,J1,J2*> by A1,TREES_4:15;
  hence thesis by FINSEQ_1:78;
end;
