I was taught to always write proofs in full.
Like for induction, in step four, i write like
If S(K) is true, S(k+1) has been proven true. Since S(1) is true, S(1+1)=S(2) is also true. Since S(2) is true, S(2+1)=S(3) is also true. Since S(3+1)=S(4) is also true and so on for all n, n >= 1...