reserve p for preProgram of SCM+FSA,
  ic for Instruction of SCM+FSA,
  i,j,k for Nat,
  fa,f for FinSeq-Location,
  a,b,da,db for Int-Location,
  la,lb for Nat;
reserve p1,p2,q for Instruction-Sequence of SCM+FSA;
reserve n for Nat;

theorem Th12:
  for t being FinPartState of SCM+FSA,p being Program of SCM+FSA,
  x being set st dom t c= Int-Locations \/ FinSeq-Locations &
  x in dom t \/ UsedI*Loc p \/ UsedILoc p
  holds x is Int-Location or x is FinSeq-Location
proof
  let t be FinPartState of SCM+FSA,p be Program of SCM+FSA,x be set;
  set D1=UsedI*Loc p;
  set D2=UsedILoc p;
  assume that
A1: dom t c= Int-Locations \/ FinSeq-Locations and
A2: x in dom t \/ D1 \/ D2;
  x in dom t \/ D1 or x in D2 by A2,XBOOLE_0:def 3;
  then
A3: x in dom t or x in D1 or x in D2 by XBOOLE_0:def 3;
  per cases by A1,A3,XBOOLE_0:def 3;
  suppose x in Int-Locations;
    hence thesis by AMI_2:def 16;
  end;
  suppose x in FinSeq-Locations;
    hence thesis by SCMFSA_2:def 5;
  end;
  suppose x in D1;
     then x in FinSeq-Locations;
    hence thesis by SCMFSA_2:def 5;
  end;
  suppose x in D2;
     then x in Int-Locations;
    hence thesis by AMI_2:def 16;
  end;
end;
