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 gt y) in Funcs(X,INT)\(b,0)) & (s.x < s.y iff g.(s, x lt 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 gt y) in Funcs(X,INT)\(b,0) iff g.(s, x gt y).b <> 0 by Th2;
  hence s.x > s.y iff g.(s, x gt y) in Funcs(X,INT)\(b,0) by Th39;
  g.(s, x lt y) in Funcs(X,INT)\(b,0) iff g.(s, x lt y).b <> 0 by Th2;
  hence thesis by Th39;
end;
