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 Th6:
  for s1,s2 being State of SCM R st IC(s1) = IC(s2) &
  (for a being Data-Location of R holds s1.a = s2.a)
    holds  s1 =  s2
proof
  let s1,s2 be State of SCM R such that
A1: IC(s1) = IC(s2);
    IC SCM R in dom s1 & IC SCM R in dom s2 by MEMSTR_0:2;
    then
A2:  s1 = DataPart s1 +* Start-At (IC s1,SCM R) &
     s2 = DataPart s2 +* Start-At (IC s2,SCM R) by MEMSTR_0:26;
  assume
A3: for a being Data-Location of R holds s1.a = s2.a;
   DataPart s1 = DataPart s2
    proof
A4:   dom DataPart s1 = Data-Locations SCM R by MEMSTR_0:9;
     hence
    dom DataPart s1 = dom DataPart s2 by MEMSTR_0:9;
     let x be object;
     assume
A5:     x in dom DataPart s1;
      then
A6:   x is Data-Location of R by A4,SCMRING2:23;
     thus (DataPart s1).x = s1.x by A5,A4,FUNCT_1:49
         .= s2.x by A6,A3
         .= (DataPart s2).x by A5,A4,FUNCT_1:49;
    end;
  hence thesis by A1,A2;
end;
