BTW, to reason a bit about this statement in a simpler fashion: "any natural number “K” can be obtained from sum of other natural numbers (less than K), where these natural numbers are powers of 2"

Think what you're doing mentally when converting a binary number to decimal. For example: 101001

Each one of those 1s represents 2^position, all added together, so:

  41 =   32 (100000 = 2^5)
       +  8 (  1000 = 2^3)
       +  1 (     1 = 2^0)

Since any natural number can be represented in decimal and binary, the rule applies that each binary 1 is the value of 2^that position, all summed up.