reserve a,b,i,j,k,l,m,n for Nat;

theorem
  for a be Real, n be non zero Nat holds
  a|^n = Sum ((a,0) Subnomial n)
  proof
    let a be Real, n be non zero Nat;
    per cases;
    suppose a is zero;
      hence thesis;
    end;
    suppose A2:a is non zero;
      a|^(n+1) - 0|^(n+1) = (a-0)*Sum ((a,0) Subnomial n) by SumS; then
      a*a|^n = a*Sum ((a,0) Subnomial n) by NEWTON:6;
      hence thesis by A2,XCMPLX_1:5;
    end;
  end;
