reserve J,J1,K for Element of Segm 13,
  b,b1,b2,c,c1,c2 for Element of SCM+FSA-Data-Loc,
  f,f1,f2 for Element of SCM+FSA-Data*-Loc;
reserve k for Nat,
  J,K,L for Element of Segm 13,
  O,P,R for Element of Segm 9;
reserve da for Int-Location,
  fa for FinSeq-Location,
  x,y for set;
reserve la,lb for Nat,
  La for Nat,
  i for Instruction of SCM+FSA,
  I for Instruction of SCM,
  l for Nat,
  LA,LB for Nat,
  dA,dB,dC,dD for Element of SCM+FSA-Data-Loc,
  DA,DB,DC for Element of SCM-Data-Loc,
  fA,fB,fC for Element of SCM+FSA-Data*-Loc,
  f,g for FinSeq-Location,
  A,B for Data-Location,
  a,b,c,db for Int-Location;
reserve S for State of SCM,
  s,s1 for State of SCM+FSA;

theorem
  for s1,s2 being State of SCM+FSA st IC s1 = IC s2 &
  (for a being Int-Location holds s1.a = s2.a) &
  (for f being FinSeq-Location holds s1.f = s2.f)
   holds s1 = s2
proof
  let s1,s2 be State of SCM+FSA such that
A1: IC(s1) = IC(s2) and
A2: for a being Int-Location holds s1.a = s2.a and
A3: for f being FinSeq-Location holds s1.f = s2.f;
  s1 in product(SCM*-VAL*SCM+FSA-OK) by CARD_3:107;
  then consider g1 being Function such that
A4: s1 = g1 and
A5: dom g1 = dom(SCM*-VAL*SCM+FSA-OK) and
  for x being object st x in dom(SCM*-VAL*SCM+FSA-OK)
   holds g1.x in (SCM*-VAL*SCM+FSA-OK).x by CARD_3:def 5;
  s2 in product(SCM*-VAL*SCM+FSA-OK) by CARD_3:107;
  then consider g2 being Function such that
A6: s2 = g2 and
A7: dom g2 = dom(SCM*-VAL*SCM+FSA-OK) and
  for x being object st x in dom(SCM*-VAL*SCM+FSA-OK)
   holds g2.x in (SCM*-VAL*SCM+FSA-OK).x by CARD_3:def 5;
A8: now
    let x be object;
    assume x in SCM+FSA-Memory;
    then x in {IC SCM+FSA} \/ SCM-Data-Loc \/ SCM+FSA-Data*-Loc by Th1;
    then
A9: x in {IC SCM+FSA} \/ SCM-Data-Loc or x in SCM+FSA-Data*-Loc
       by XBOOLE_0:def 3;
A10:   Int-Locations = SCM+FSA-Data-Loc;
    per cases by A9,XBOOLE_0:def 3;
    suppose
      x in {IC SCM+FSA};
      then x = IC SCM+FSA by TARSKI:def 1;
      hence g1.x = g2.x by A1,A4,A6;
    end;
    suppose
      x in SCM-Data-Loc;
      then x is Int-Location by A10,AMI_2:def 16;
      hence g1.x = g2.x by A2,A4,A6;
    end;
    suppose
      x in SCM+FSA-Data*-Loc;
      then x is FinSeq-Location by Def3;
      hence g1.x = g2.x by A3,A4,A6;
    end;
  end;
  thus thesis by A4,A5,A6,A7,A8;
end;
