“In the case” means “if”.

For example:

In the case the Lessee fails to undertake reparations within this period, the Lessor may terminate the Lease. = If the Lessee fails to undertake reparations…

You may also use this phrase with “that”:

In the case that the Lessee fails to undertake reparations…

“In case of” (without “the”) also means “if”. It is used exclusively in formal notices and announcements.

For example:

In case of fire do not use the lifts. = If there is a fire…

In case of rain the wedding reception will be held in the hotel ballroom. = If it rains…

“In case of” (without “the”) SHOULD NOT be used except in formal notices and announcements.

“In the case of” means “as regards” / “with regard to” / “as far as … is concerned” etc.

For example:

In the case of the Property, the lease agreement has been executed for a period of ten years. = As regards the Property…

“In the case of” can also mean “for”, as in this example:

The authority may impose a fine in the case of non-compliance with the regulations. = …for non-compliance…

“In case” is different. It is an idiomatic phrase that we use when we talk about taking precautions.

For example:

In case it rains I will take an umbrella.

This means: Because there is a possibility that it will rain later, I will take an umbrella now.

“Just in case…” is a common informal version of the phrase.

“In case” DOES NOT mean “in the case” / “if”

More examples:

In case my house burns down I’ll take out insurance. (BEFORE my house burns down, as a precaution)
In the case my house burns down I’ll take out insurance. (AFTER my house burns down. That’s too late!)

In case I have a car accident I’ll wear a seatbelt. (BEFORE I have an accident, as a precaution)
In the case I have a car accident I’ll wear a seatbelt. (AFTER the accident. That’s too late!)

Now consider this example:

In case the Lessee does not pay the Rent within three days of the Lessor’s final request, the Lessor is entitled to terminate the Agreement without notice.

This means: Because there is a possibility that the Lessee will not pay the Rent within three days of the Lessor’s final request, the Lessor is entitled to terminate the Agreement without notice BEFORE the Lessee’s non-payment, as a precaution.

The Lessee would be unlikely to agree to this clause.

Now add “the”:

In the case the Lessee does not pay the Rent within three days of the Lessor’s final request, the Lessor is entitled to terminate the Agreement without notice.

This means: If the Lessee does not pay the Rent within three days of the Lessor’s final request, the Lessor is entitled to terminate the Agreement without notice.

This is a standard conditional sentence (and a standard contractual clause).

Summary