reserve a, b, c, a1, a2, b1, b2 for Int-Location,
  l, l1, l2 for Nat,
  f, g, f1, f2 for FinSeq-Location,
  i, j for Instruction of SCM+FSA,
  X, Y for set;
reserve p, r for preProgram of SCM+FSA,
  I, J for Program of SCM+FSA,
  k, m, n for Nat;
reserve L for finite Subset of Int-Locations;
reserve L for finite Subset of FinSeq-Locations;
reserve s, t for State of SCM+FSA;
reserve P for Instruction-Sequence of SCM+FSA;

theorem Th62:
  not f in UsedInt*Loc i implies Exec(i, s).f = s.f
proof
  assume
A1: not f in UsedInt*Loc i;
 InsCode i <= 12 by SCMFSA_2:16;
  then InsCode i = 0 or ... or InsCode i = 12;
  then per cases;
  suppose
    InsCode i = 0;
    then i = halt SCM+FSA by SCMFSA_2:95;
    hence thesis by EXTPRO_1:def 3;
  end;
  suppose
    InsCode i = 1;
    then ex a, b st i = a:=b by SCMFSA_2:30;
    hence thesis by SCMFSA_2:63;
  end;
  suppose
    InsCode i = 2;
    then ex a, b st i = AddTo(a,b) by SCMFSA_2:31;
    hence thesis by SCMFSA_2:64;
  end;
  suppose
    InsCode i = 3;
    then ex a, b st i = SubFrom(a, b) by SCMFSA_2:32;
    hence thesis by SCMFSA_2:65;
  end;
  suppose
    InsCode i = 4;
    then ex a, b st i = MultBy(a, b) by SCMFSA_2:33;
    hence thesis by SCMFSA_2:66;
  end;
  suppose
    InsCode i = 5;
    then ex a, b st i = Divide(a, b) by SCMFSA_2:34;
    hence thesis by SCMFSA_2:67;
  end;
  suppose
    InsCode i = 6;
    then ex l st i = goto l by SCMFSA_2:35;
    hence thesis by SCMFSA_2:69;
  end;
  suppose
    InsCode i = 7;
    then ex l, a st i = a=0_goto l by SCMFSA_2:36;
    hence thesis by SCMFSA_2:70;
  end;
  suppose
    InsCode i = 8;
    then ex l, a st i = a>0_goto l by SCMFSA_2:37;
    hence thesis by SCMFSA_2:71;
  end;
  suppose
    InsCode i = 9;
    then ex a, b, g st i = b:=(g,a) by SCMFSA_2:38;
    hence thesis by SCMFSA_2:72;
  end;
  suppose
    InsCode i = 10;
    then consider a, b, g such that
A2: i = (g,a):=b by SCMFSA_2:39;
    UsedInt*Loc i = {g} by A2,Th33;
    then f <> g by A1,TARSKI:def 1;
    hence thesis by A2,SCMFSA_2:73;
  end;
  suppose
    InsCode i = 11;
    then ex a, g st i = a:=len g by SCMFSA_2:40;
    hence thesis by SCMFSA_2:74;
  end;
  suppose
    InsCode i = 12;
    then consider a,g such that
A3: i = g:=<0,...,0>a by SCMFSA_2:41;
    UsedInt*Loc i = {g} by A3,Th34;
    then f <> g by A1,TARSKI:def 1;
    hence thesis by A3,SCMFSA_2:75;
  end;
end;
