reserve i for Nat, x,y for set;
reserve S for non empty non void ManySortedSign;
reserve X for non-empty ManySortedSet of S;

theorem
  for U1,U2 be preIfWhileAlgebra st the UAStr of U1 = the UAStr of U2
  holds EmptyIns U1 = EmptyIns U2 &
  for I1,J1 being Element of U1
  for I2,J2 being Element of U2 st I1 = I2 & J1 = J2
  holds I1\;J1 = I2\;J2 & while(I1,J1) = while(I2,J2) &
  for C1 being Element of U1
  for C2 being Element of U2 st C1 = C2
  holds if-then-else(C1,I1,J1) = if-then-else(C2,I2,J2);
