Overriden class member annotated with a subtype could be a recipe for disaster (by design.. ;) )

TypeScript Version:
1.9.0-dev.20160505 with both strictNullChecks and noImplicitAny enabled.

Code:

class Animal {
    name: string = "Any animal";
}

class Dog extends Animal {
    name: string = "Dog";

    bark() {
        console.log("bark");
    }
}

class AnimalCage {
    member: Animal = new Animal();
}

class DogCage extends AnimalCage {
    member: Dog;
}

let dogCage = new DogCage();

if (dogCage.member != null) { // Yup, it is even checked for null or undefined, 
                              // but it turned out that wasn't enough..
    dogCage.member.bark(); // Bang!
}

Expected behavior:
Design a language where inheritance actually works.
(Edit: I was felling very 'passionate' when I wrote this.. sorry :) nothing personal)

Actual behavior:
By design:

    dogCage.member.bark();
                   ^

TypeError: dogCage.member.bark is not a function
    at Object.<anonymous> (C:\X\Documents\Drafts\TypeScript\Code\a.js:38:20)
    at Module._compile (module.js:541:32)
    at Object.Module._extensions..js (module.js:550:10)
    at Module.load (module.js:456:32)
    at tryModuleLoad (module.js:415:12)
    at Function.Module._load (module.js:407:3)
    at Function.Module.runMain (module.js:575:10)
    at startup (node.js:159:18)
    at node.js:444:3

Resolution:
tslint volunteers, for the rescue!

Start by compiling and running the TypeScript example in the issue, then read the discussion about the intended inheritance behavior. A viable contribution would need an agreed language design, an implementation plan, and regression coverage showing the resulting behavior.