When the call is at or out of the money, the result is obvious: the call will have some value but the difference between stock price and strike is nonpositive.
Consider the case that the option is in the money and its current price $C$ is lower than the difference between stock price $S$ and the strike $K$, in symbols: $C < S-K$. If I own the stock, I could sell it to buy the option and lent out $K$ until expiration, at the end of the period I either get $K$ if the option is worth nothing or $S$ is the option is in the money. So by selling the stock for the option and lending out the difference: I've put a floor on the value of my position and pocketed $S - K - C$. This would be a free lunch.
As @BabaYaga points out, the statement and my answer above only hold in the absence of dividends. The statement can be fixed by including the dividend in the difference. Put-call parity before expiration is $$C - P = S - K - D$$ where $P$ is the price of a put with the same strike and $D$ the dividend expected between now and maturity. The value of the call is reduced by the dividends. If the dividends are sufficiently large, the original statement would be false.
the Ask Price of a Call Option is always higher than the difference between the Strike Price and the price of underlying stock [more precisely, the price of the underlying minus the strike].
This is definitely true for American call options, that can be exercised immediately (as well as all the way to maturity): roughly speaking, you can "indirectly" purchase the underlying asset by purchasing an american call and paying the strike price to exercise the option-- thus the price of the asset can't exceed the price of the option plus its strike price, or there would be significant arbitrage opportunities.
I'd say it is not necessarily true for European calls, that can be exercised only at maturity. For example, consider a very, very stable utility stock, that is currently priced at 1000 EUR, and will pay a 50 EUR dividend in 4 days from now; and a European call option on the stock with a strike of 1 EUR maturing in a week from now. The call option price is then very close to 950 EUR, so it's still quite a bit below the current stock price (1000 EUR) minus the strike (1 EUR), violating the statement above.