reserve A for preIfWhileAlgebra;
reserve A for Euclidean preIfWhileAlgebra;
reserve X for non empty countable set;
reserve T for Subset of Funcs(X, INT);
reserve f for Euclidean ExecutionFunction of A, Funcs(X, INT), T;
reserve A for Euclidean preIfWhileAlgebra,
  X for non empty countable set,
   z for (Element of X),
  s,s9 for (Element of Funcs(X, INT)),
  T for Subset of Funcs(X, INT),
  f for Euclidean ExecutionFunction of A, Funcs(X, INT), T,
  v for INT-Variable of A,f,
  t for INT-Expression of A,f;
reserve i for Integer;

theorem Th31:
  for x being Variable of f holds f.(s, x*=i).x = s.x*i & for z st
  z <> x holds f.(s, x*=i).z = s.z
proof
  let x be Variable of f;
A2: .(^(x qua Element of X)).s = s.((^(x qua Element of X)).s) by Def19;
  (.x*i).s = ((.x).s*i) by Def10;
  hence thesis by A2,Th24;
end;
