reserve x,y for set;
reserve i, j, k for Nat;
reserve I,J,K for Element of Segm 9,
  a,a1 for Nat,
  b,b1,c for Element of Data-Locations SCM;
reserve a, b for Data-Location,
  loc for Nat;
reserve I,J,K for Element of Segm 9,
  a,a1 for Nat,
  b,b1,c for Element of Data-Locations SCM,
  da,db for Data-Location;

theorem
  for s1,s2 being State of SCM st IC(s1) = IC(s2) &
  (for a being Data-Location holds s1.a = s2.a)
    holds  s1 =  s2
proof
  let s1,s2 be State of SCM such that
A1: IC(s1) = IC(s2);
    IC SCM in dom s1 & IC SCM in dom s2 by MEMSTR_0:2;
    then
A2:  s1 = DataPart s1 +* Start-At (IC s1,SCM) &
     s2 = DataPart s2 +* Start-At (IC s2,SCM) by MEMSTR_0:26;
  assume
A3: for a being Data-Location holds s1.a = s2.a;
   DataPart s1 = DataPart s2
    proof
A4:   dom DataPart s1 = Data-Locations SCM 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 by A4,AMI_2:def 16,AMI_3:27;
     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;
