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 Th50:
  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 (
t.s is odd iff g.(s, t is_odd) in Funcs(X,INT)\(b,0)) & (t.s is even iff g.(s,
  t is_even) in Funcs(X,INT)\(b,0))
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;
A1: (t.s) mod 2 = 0 or (t.s) mod 2 = 1 by PRE_FF:6;
A2: t.s = ((t.s) div 2)*2 + ((t.s) mod 2) by INT_1:59;
  f.(s, t is_odd).b = (t.s) mod 2 by Th48;
  hence t.s is odd iff f.(s, t is_odd) in Funcs(X,INT)\(b,0) by A1,A2,Th2;
A3: (t.s+1) mod 2 = 0 or (t.s+1) mod 2 = 1 by PRE_FF:6;
A4: t.s+1 = ((t.s+1) div 2)*2 + ((t.s+1) mod 2) by INT_1:59;
  f.(s, t is_even).b = (t.s+1) mod 2 by Th48;
  hence thesis by A3,A4,Th2;
end;
