reserve l, m, n for Nat,
  i,j,k for Instruction of SCMPDS,
  I,J,K for Program of SCMPDS,
  p,q,r for PartState of SCMPDS;
reserve a,b,c for Int_position,
  s,s1,s2 for State of SCMPDS,
  k1,k2 for Integer;

theorem Th3:
  for k1,k2 be Nat st k1 <> k2 holds DataLoc(k1,0) <>
  DataLoc(k2,0)
proof
  let k1,k2 be Nat;
  assume
A1: k1<>k2;
  assume
  DataLoc(k1,0) = DataLoc(k2,0);
  then |.k1+0.| =|.k2+0.| by XTUPLE_0:1;
  then k1=|.k2.| by ABSVALUE:def 1;
  hence contradiction by A1,ABSVALUE:def 1;
end;
