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;

theorem Th26:
  UsedILoc I = UsedILoc Directed I
proof
 set A = { UsedIntLoc i : i in rng I },
     B = { UsedIntLoc i : i in rng Directed I };
A1: A c= B
   proof let e be object;
    assume e in A;
     then consider i such that
A2:   e = UsedIntLoc i and
A3:   i in rng I;
     consider x being object such that
A4:   x in dom I and
A5:   I.x = i by A3,FUNCT_1:def 3;
     set j = (Directed I).x;
   x in dom Directed I by A4,FUNCT_4:99;
     then
A6:    j in rng Directed I by FUNCT_1:3;
     reconsider j as Instruction of SCM+FSA by A6;
     now per cases;
      suppose
A7:      i = halt SCM+FSA;
       then j = goto card I by A4,A5,FUNCT_4:106;
      hence UsedIntLoc i = UsedIntLoc j by Th15,A7,Th13;
      end;
      suppose i <> halt SCM+FSA;
      hence UsedIntLoc i = UsedIntLoc j by A5,FUNCT_4:105;
      end;
     end;
    hence e in B by A2,A6;
   end;
  B c= A
   proof let e be object;
    assume e in B;
     then consider i such that
A8:   e = UsedIntLoc i and
A9:   i in rng Directed I;
     consider x being object such that
A10:   x in dom Directed I and
A11:   (Directed I).x = i by A9,FUNCT_1:def 3;
     set j = I.x;
A12:    x in dom I by A10,FUNCT_4:99;
     then
A13:    j in rng I by FUNCT_1:3;
     reconsider j as Instruction of SCM+FSA by A13;
     now per cases;
      suppose
A14:      j = halt SCM+FSA;
       then
      i = goto card I by A11,FUNCT_4:106,A12;
      hence UsedIntLoc i = UsedIntLoc j by A14,Th13,Th15;
      end;
      suppose j <> halt SCM+FSA;
      hence UsedIntLoc i = UsedIntLoc j by A11,FUNCT_4:105;
      end;
     end;
    hence e in A by A8,A13;
   end;
 hence thesis by A1,XBOOLE_0:def 10;
end;
