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

theorem
  for a be non negative Real holds
    Product ((a,1) Subnomial n) = a|^((n+1) choose 2)
  proof
    let a be non negative Real;
    set l = a|^(n choose 2),f = ((a,1) Subnomial n),g = ((1,a) Subnomial n);
    A1: ((n+1) choose 2)*2 = (n*(n+1))/2*2 by NUMPOLY1:72
    .= n*(n+1);
    A2: (a|^((n+1) choose 2))|^2 = a|^(((n+1) choose 2)*2) by NEWTON:9
    .= Product ((a*1,a*1) Subnomial n) by A1,PRA
    .= Product (f(#)g) by ST
    .= (Product f)*(Product g) by PRN
    .= (Product f)*(Product f)
      by SFE,RVSUM_3:4
    .= (Product f)|^2 by NEWTON:81;
    a|^((n+1) choose 2) = sqrt((Product f)|^2) by A2
    .= Product f;
    hence thesis;
  end;
