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

theorem
  for a,b be Real, n,i be Nat holds
    ((a,b) Subnomial n).i <= ((|.a.|,|.b.|)Subnomial n).i
  proof
    let a,b be Real, n,i be Nat;
    reconsider f = ((|.a.|,|.b.|) Subnomial n)
      as nonnegative-yielding FinSequence of REAL;
    per cases;
    suppose not i in dom ((a,b) Subnomial n);
      hence thesis by FUNCT_1:def 2;
    end;
    suppose
      A0: i in dom ((a,b) Subnomial n); then
      A1: 1 <= i  <= len ((a,b) Subnomial (n+1-1)) by FINSEQ_3:25; then
      reconsider l = i-1 as Nat;
      ex k be Nat st n+1 = (l+1) + k by A1,NAT_1:10; then
      reconsider k = (n+1) - (l+1) as Nat;
      A2: k = n - l & l = i - 1;
      A3: |.a.||^k = |.a|^k.| & |.b.||^l = |.b|^l.| by TAYLOR_2:1;
      A4: dom ((a,b) Subnomial n) = dom (Newton_Coeff n) by DOMN
      .= dom ((|.a.|,|.b.|)Subnomial n) by DOMN;
      |.(a|^k)*(b|^l).| >= (a|^k)*(b|^l) by ABSVALUE:4; then
      (|.a|^k.|) * (|.b|^l.|) >= a|^k*b|^l by COMPLEX1:65; then
      (|.a.||^k)*(|.b.||^l) >= ((a,b) Subnomial (l+k)).(l+1) by A0,A2,A3,LmS1;
      hence thesis by A0,A2,A4,LmS1;
    end;
  end;
