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

theorem
  for a be Real, n be Nat, i be non zero Nat holds
    ((a,-a) Subnomial (n+2*i)).(2*i) = -a|^(n+2*i)
  proof
    let a be Real, n be Nat, i be non zero Nat;
    A1: len ((a,a) Subnomial ((n+2*i)+1-1)) = n+2*i+1 &
     len ((a,-a) Subnomial ((n+2*i)+1-1)) = n+2*i+1;
    2*i >= 1 & 2*i+(n+1) >= 2*i + 0 by XREAL_1:6,NAT_1:14; then
    A2: 2*i in dom ((a,a) Subnomial (2*i+n)) &
    2*i in dom ((a,-a) Subnomial (2*i+n)) by A1,FINSEQ_3:25; then
    ((a*1,(-1)*a) Subnomial (n+2*i)).(2*i) =
    ((a,a) Subnomial (n+2*i)).(2*i)*((1,-1) Subnomial (n+2*i)).(2*i) by STT
    .= a|^(n+2*i)*(-1) by A2,CONST;
    hence thesis;
  end;
