reserve i, j, k for Nat,
  n for Nat,
  IL for non empty set,
  N for with_non-empty_elements set;
reserve R for non trivial Ring,
  a, b for Data-Location of R,
  loc for Nat,
  I for Instruction of SCM R,
  p for FinPartState of SCM R,
  s, s1, s2 for State of SCM R,
  P,P1,P2 for Instruction-Sequence of SCM R,
  q for FinPartState of SCM;

theorem
  for i,j being Nat holds i <> j implies dl.(R,i) <> dl.(R,j)
proof
  let i,j be Nat;
  assume
A1: i <> j;
  dl.(R,j) = [1,j] & dl.(R,i) = [1,i] by Th1;
  hence thesis by A1,XTUPLE_0:1;
end;
