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 Th60:
  not c in UsedIntLoc i implies Exec(i, s).c = s.c
proof
  assume
A1: not c in UsedIntLoc 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 consider a, b such that
A2: i = a:=b by SCMFSA_2:30;
    UsedIntLoc i = {a, b} by A2,Th14;
    then c <> a by A1,TARSKI:def 2;
    hence thesis by A2,SCMFSA_2:63;
  end;
  suppose
    InsCode i = 2;
    then consider a, b such that
A3: i = AddTo(a,b) by SCMFSA_2:31;
    UsedIntLoc i = {a, b} by A3,Th14;
    then c <> a by A1,TARSKI:def 2;
    hence thesis by A3,SCMFSA_2:64;
  end;
  suppose
    InsCode i = 3;
    then consider a, b such that
A4: i = SubFrom(a, b) by SCMFSA_2:32;
    UsedIntLoc i = {a, b} by A4,Th14;
    then c <> a by A1,TARSKI:def 2;
    hence thesis by A4,SCMFSA_2:65;
  end;
  suppose
    InsCode i = 4;
    then consider a, b such that
A5: i = MultBy(a, b) by SCMFSA_2:33;
    UsedIntLoc i = {a, b} by A5,Th14;
    then c <> a by A1,TARSKI:def 2;
    hence thesis by A5,SCMFSA_2:66;
  end;
  suppose
    InsCode i = 5;
    then consider a, b such that
A6: i = Divide(a, b) by SCMFSA_2:34;
    UsedIntLoc i = {a, b} by A6,Th14;
    then c <> a & c <> b by A1,TARSKI:def 2;
    hence thesis by A6,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 consider a, b, f such that
A7: i = b:=(f,a) by SCMFSA_2:38;
    UsedIntLoc i = {a, b} by A7,Th17;
    then c <> b by A1,TARSKI:def 2;
    hence thesis by A7,SCMFSA_2:72;
  end;
  suppose
    InsCode i = 10;
    then ex a, b, f st i = (f,a):=b by SCMFSA_2:39;
    hence thesis by SCMFSA_2:73;
  end;
  suppose
    InsCode i = 11;
    then consider a, f such that
A8: i = a:=len f by SCMFSA_2:40;
    UsedIntLoc i = {a} by A8,Th18;
    then c <> a by A1,TARSKI:def 1;
    hence thesis by A8,SCMFSA_2:74;
  end;
  suppose
    InsCode i = 12;
    then ex a,f st i = f:=<0,...,0>a by SCMFSA_2:41;
    hence thesis by SCMFSA_2:75;
  end;
end;
