reserve a,b,c,d,m,x,n,j,k,l for Nat,
  t,u,v,z for Integer,
  f,F for FinSequence of NAT;
reserve p,q,r,s for real number;

theorem Th55: :: see SERIES_3:6
  a<>b implies 2*a*b < a|^2 + b|^2
  proof
    A0: a*a = a|^2 & b*b = b|^2 by NEWTON:81;
    assume a <> b; then
    (a - b) is non zero real & 2 is even; then
    A1: (a - b)|^2 > 0;
    assume not thesis; then
    2*a*b -2*a*b >= a|^2 + b|^2 - 2*a*b by XREAL_1:9; then
    0 >= (a-b)*(a-b) by A0;
    hence contradiction by A1,NEWTON:81;
  end;
