Lithium fat for bicycle chain? In the local hardware store I found this ultra cheap lithium soap fat. Can I use this for lubrication of my bicycle chain?
No, lithium grease or similar grease compounds are in general not suitable for bike chains.

Bike chains need lubrication of inner surfaces between rollers. Chain lubes contain solvents to reduce viscosity enough to creep into these gaps. The solvent dissipates and leaves a high viscosity oil.

The surface does not need extra lubrication. It is best to remove all chain lube from it to prevent dust and grit adhering to the chain. Eventually it would get into the chain rollers and increase wear.

Lithium complex grease is good for mounting parts. For example seat posts or screw threads. It can serve as an allround grease to server the purpose of anti seize compounds and bearing grease. There are a few components, like suspension forks, that must not be lubricated with lithium grease though.