reserve l, m, n for Nat,
  i,j,k for Instruction of SCM+FSA,
  I,J,K for Program of SCM+FSA;
reserve a,b for Int-Location,
  f for FinSeq-Location,
  s,s1,s2 for State of SCM+FSA;

theorem Th3:
  InsCode i in {0,6,7,8} or Exec(i,s).IC SCM+FSA = IC s + 1
proof
  assume
A1: not InsCode i in {0,6,7,8};
  then
A2: InsCode i <> 0 & InsCode i <> 6 by ENUMSET1:def 2;
A3: InsCode i <> 7 & InsCode i <> 8 by A1,ENUMSET1:def 2;
  InsCode i = 0 or ... or InsCode i = 12 by SCMFSA_2:16;
  then per cases by A2,A3;
  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 = 9;
    then ex a,b,f st i = b:=(f,a) by SCMFSA_2:38;
    hence thesis by 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 ex a,f st i = a:=len f by SCMFSA_2:40;
    hence thesis by 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;
