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 Th48:
  for b being Element of X for g being Euclidean ExecutionFunction
of A,Funcs(X,INT), Funcs(X,INT)\(b,0) for t being INT-Expression of A,g holds g
.(s, t is_odd).b = (t.s) mod 2 & g.(s, t is_even).b = (t.s+1) mod 2 & for z st
  z <> b holds g.(s, t is_odd).z = s.z & g.(s, t is_even).z = s.z
proof
  let b be Element of X;
  let f be Euclidean ExecutionFunction of A,Funcs(X,INT), Funcs(X,INT)\(b,0);
  let t be INT-Expression of A,f;
  reconsider y = b as Variable of f by Def2;
A1: t is_odd = y:=(t mod .(2,A,f));
  dom(t+1) = Funcs(X,INT) by FUNCT_2:def 1;
  then
A2: (t+1).s = 1+(t.s) by VALUED_1:def 2;
A3: .(2,A,f).s = 2;
  then
A4: ((t+1) mod .(2,A,f)).s = ((t+1).s) mod 2 by Def30;
  (t mod .(2,A,f)).s = (t.s) mod 2 by A3,Def30;
  hence thesis by A1,A2,A4,Th26;
end;
