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 Th29:
  for x being Variable of f for t being INT-Expression of A,f
  holds f.(s, x+=t).x = s.x+t.s & for z st z <> x holds f.(s, x+=t).z = s.z
proof
  let x be Variable of f;
  let t be INT-Expression of A,f;
A1: (^x).s = x;
  dom (.x+t) = Funcs(X, INT) by FUNCT_2:def 1;
  then
A2: (.x+t).s = (.x).s+t.s by VALUED_1:def 1;
  (.x).s = s.x by Th22;
  hence thesis by A1,A2,Th24;
end;
