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;
