When I added the lattice object, confirm it with the borders of chest and use Lattice modifier the scale of chest object has changed (I scrolled the strength of modifier and near about zero it come back to normal size).
Also I tried to apply the transformations (scale of modified object), but it comes weird and object go through the borders of lattice object.
Question: did you by any chance make the (very common and natural) mistake and scaled the Lattice in Edit Mode?
Remove the Lattice you have and add a new one and scale it in Object Mode to fit around the chest.
It works) thanks you) but one little question - if I applied the scale of Lattice (so it has 1,1,1) when it hasn't any objects with Lattice modifier, why it doesn't work?
Any way, I appreciate your answer, thanks)
A lattice should be scaled in Object Mode only. That means that its Scale and Dimensions are the same (unless you are using Imperial units...). Applying the Scale of the Lattice makes its Scale (1, 1, 1) while its Dimension don't change. That would be the same as scaling in Edit Mode and that is what we didn't want to do with Lattices.
Why it has to be like this I don't know unfortunately.