theorem
  for b being Element of X for g being Euclidean ExecutionFunction of A,
Funcs(X,INT), Funcs(X,INT)\(b,0) for x,y being Variable of g holds (s.x <= s.y
  iff g.(s, x leq y) in Funcs(X,INT)\(b,0)) & (s.x >= s.y iff g.(s, x geq y) in
  Funcs(X,INT)\(b,0))
proof
  let b be Element of X;
  let g be Euclidean ExecutionFunction of A,Funcs(X,INT), Funcs(X,INT)\(b,0);
  let x,y be Variable of g;
  g.(s, x leq y) in Funcs(X,INT)\(b,0) iff g.(s, x leq y).b <> 0 by Th2;
  hence s.x <= s.y iff g.(s, x leq y) in Funcs(X,INT)\(b,0) by Th35;
  g.(s, x geq y) in Funcs(X,INT)\(b,0) iff g.(s, x geq y).b <> 0 by Th2;
  hence thesis by Th35;
end;
