Below is a simple proof using predicate logic.

This is a very good example to show you that not all proofs can be proved backwards, sometimes we also need to work forwards. Working backwards as shown, the Not Introduction rule is used (reason explained in the propositional logic proof). This generates a box with ∃t [ ∃u [ R(t,u) ∧ T ∧ ¬ ( ∃v [ R(t,v) ∧ R(u,v) ] ) ] ] as an assumption and bottom as the goal. Then Exists Elimination is used forwards, this is because we need constants to give values for the variables t and u in ∀t [ ∀u [ R(t,u) → ∃v [ R(t,v) ∧ R(u,v) ] ] ] and since the assumption is an exists formula it tells us that there is a value for t and u. So Exists Elimination is used forwards and a bottom goal is entered when prompted. You can either eliminate all or one at a time. In this case both are eliminated as shown below.This in essence creates a box with two new constants and replaces t and u in the formula.

Then the And Elimination rule is used forwards on the ground formula derived to obtain the sub formula which matches the sub formula of given, you can either eliminate all or skip, in the case shown below eliminate all is chosen, skip can be chosen to eliminate a specific sub formula, but in our case we need the remaining sub formulas hence the choice. Now For All Arrow Elimination rule is used forwards by clicking R(sk1,sk2) and the given, this generates a new proof line which is the converse of the one of the formulas in the box. How is this done? Our system is sophisticated enough to match the left side of the given to R(sk1,sk2) and unifies t to sk1 and u to sk2 and replaces all occurrences of t and u in the for all formula and the right sub formula with the replaced constants is written as the new proof line. You could also do the above by using For All Elimination first and then Arrow Elimination.

Now use the Not elimination rule forwards and click on the newly derived line and its converse. This gives you bottom as the new goal. Then use the tick rule backwards and click on the derived bottom, thereby closing the current box. Then continue using the tick rule backwards until the proof is completed.

Note: When a rule is said to be used forwards it always means to click both the empty line and the rule; when used backwards it means click both the goal line and the rule.