Allen (1983) remarks that the symbolic constraint (IA {f fi} IC) cannot hold given the existing constraints between IA, IB, IC and ID. In the labeled point-based TCN, (IA {f fi} IC) is represented by a set of constraints among ending points of IA and IC. Moreover, the labels associated to each labeled elemental constraint allow us to determine whether a set of elemental constraints between different pairs of time-points can be part of a global solution (*Theorem 10*). Thus, we can deduce whether (IA {f fi} IC) can hold in the point-based TCN.

The existing constraints between the ending time-points of IC and IA, with their associated labelsets are:
IC
+{(-∞ ∞){R25 R30 R29 R17 R22 R21 R0)∨{R27 R28 R29 R19 R20 R21 R0},
 (-∞ 0){R27 R28 R29 R17 R22 R21 R9 R10 R15 R16 R1 R2 R7 R0 R8},
 (0 ∞){R25 R30 R29 R19 R20 R21 R11 R12 R13 R14 R3 R4 R5 R0 R6}} IA
+
IC
-{(0 ∞){R27 R28 R29 R17 R22 R21 R9 R10 R15 R16 R1 R2 R7 R8 R0},
 [0 0]{R25 R30 R29 R17 R22 R21 R0}∨{R27 R28 R29 R19 R20 R21 R0},
 (-∞ 0){R25 R30 R29 R19 R20 R21 R11 R12 R13 R14 R3 R4 R5 R6 R0}} IA
-
Let's ask for each disjunct in (IA {f fi} IC):
i) The constraint (IA {f} IC) implies (IC
+{[0 0]} IA
+) ∧ (IC
-{(-∞ 0)} IA
-). According to Theorem 10, these constraints hold iff the set-union of the label sets associated to (IC
+[0 0]

IA
+) and to (IC
-(-∞ 0) IA
-) is not an I-L-Set. We have two possibilities:
i.1) {R25 R30 R29 R17 R22 R21 R0} ∪ {R25 R30 R29 R19 R20 R21 R11 R12 R13 R14 R3 R4 R5 R6 R0} =
{R6 R5 R4 R3 R20 R19 R25 R30 R29 R17 R22 R21 R11 R12 R13 R14 R0 }, or i.2) {R27 R28 R29 R19 R20 R21 R0} ∪ {R25 R30 R29 R19 R20 R21 R11 R12 R13 R14 R3 R4 R5 R6 R0} =
{R14 R13 R12 R11 R30 R25 R27 R28 R29 R19 R20 R21 R3 R4 R5 R0 R6}.
However, both label sets (i.1, i.2) are I-L-Sets: For instance, {R19 R22} and {R27 R30} are I-LSets (Table 2) and they are subsets of i.1 and i.2, respectively. Thus, (IA {f} IC) does not hold.

ii) The constraint (IA {fi} IC) implies (IC
+{[0 0]} IA
+) ∧ (IC
-{ (0 ∞)} IA
-). Similarly:
ii.1) {R25 R30 R29 R17 R22 R21 R0} ∪ {R27 R28 R29 R17 R22 R21 R9 R10 R15 R16 R1 R2 R7 R8 R0} =
{R16 R15 R10 R9 R28 R27 R25 R30 R29 R17 R22 R21 R1 R2 R7 R0 R8}.

This label set is an I-L-Set. For instance, {R30 R27} is an I-L-Set. Also, ii.2) {R27 R28 R29 R19 R20 R21 R0} ∪ {R27 R28 R29 R17 R22 R21 R9 R10 R15 R16 R1 R2 R7 R8 R0} =
{R8 R7 R2 R1 R22 R17 R27 R28 R29 R19 R20 R21 R9 R10 R15 R16 R0}.
Both these label sets (ii.1, *ii.2*) are also I-L-Sets. For instance, {R30 R27} and {R19 R22} are I-LSets. Thus, (IA {fi} IC) does not hold either.

In conclusion, the symbolic constraint (IA {f fi} IC) cannot hold on the globally labeled-consistent point-based TCN. This conclusion could be also obtained from a minimal IA-TCN (Figure 11.b).

Additionally, we have that (IA {f fi} IC) implies (IA
+[0 0] IC
+). That is, if the constraint (IA
+[0 0]
IC
+) holds, we have that the associated constraints to the label sets {R25 R30 R29 R17 R22 R21 R0} or
{R27 R28 R29 R19 R20 R21 R0} should also hold. Each one of these label sets implies (IC
- {[0 0]} IA
-).

That is: (IA
+[0 0] IC
+) → (IC
- {[0 0]} IA
-). Thus, the only way that (IA
+[0 0] IC
+) can hold is if (IA
{e} IC) holds. These relations will be detailed in Section 6.