Thus, between events E0 and E1, A advances by $\color {ForestGreen}{t_{1}}$ and B by $\color {Blue}{t'_{1}=at_{1}}$ by (1). Therefore

${\frac {\color {ForestGreen}{\text{rate of A}}}{\color {Blue}{\text{rate of B}}}}={\frac {\color {ForestGreen}{t_{1}}}{\color {Blue}{at_{1}}}}={\frac {1}{a}}>1\qquad {\text{(3)}}$

...

Thus, between events E0 and E2, B advances by $\color {Brown}{t'_{2}}$ and A by $\color {Red}{t_{2}=at'_{2}}$ by (2). Therefore

${\frac {\color {Red}{\text{rate of A}}}{\color {Brown}{\text{rate of B}}}}={\frac {\color {Red}{at'_{2}}}{\color {Brown}{t'_{2}}}}=a<1\qquad {\text{(4)}}$

Equations (3) and (4) are contradictory: hence the theory requiring them must be false.

(end quote)

Dingle should have written as follows:

(start correction)

Thus, between events E0 and E1, A, which is not present at both events, advances by $\color {ForestGreen}{t_{1}}$ and B, which is present at both events, by $\color {Blue}{t'_{1}=at_{1}}$ by (1). Therefore

${\frac {\color {ForestGreen}{\text{rate of clock not present at both events E0 and E1}}}{\color {Blue}{\text{rate of clock present at both events E0 and E1}}}}={\frac {\color {ForestGreen}{\text{coordinate time of E1}}}{\color {Blue}{\text{proper time of E1}}}}={\frac {\color {ForestGreen}{\text{rate of A}}}{\color {Blue}{\text{rate of B}}}}={\frac {\color {ForestGreen}{t_{1}}}{\color {Blue}{t'_{1}}}}={\frac {\color {ForestGreen}{t_{1}}}{\color {Blue}{at_{1}}}}={\frac {1}{a}}>1\qquad {\text{(3)}}$

...

Thus, between events E0 and E2, B, which is not present at both events, advances by $\color {Brown}{t'_{2}}$ and A, which is present at both events, by $\color {Red}{t_{2}=at'_{2}}$ by (2). Therefore

${\frac {\color {Brown}{\text{rate of clock not present at both events E0 and E2}}}{\color {Red}{\text{rate of clock present at both events E0 and E2}}}}={\frac {\color {Brown}{\text{coordinate time of E2}}}{\color {Red}{\text{proper time of E2}}}}={\frac {\color {Brown}{\text{rate of B}}}{\color {Red}{\text{rate of A}}}}={\frac {\color {Brown}{t'_{2}}}{\color {Red}{t_{2}}}}={\frac {\color {Brown}{t'_{2}}}{\color {Red}{at'_{2}}}}={\frac {1}{a}}>1\qquad {\text{(4)}}$

Equations (3) and (4) are consistent and say that any event's coordinate time is always larger than its proper time:

hence there is no reason to say that the theory requiring them must be false.