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

theorem NYS1:
  for f,g be real-valued FinSequence st
    (for x be Nat holds f.x >= g.x & ex i st f.(i+1) > g.(i+1)) holds
      Sum f > Sum g
  proof
    let f,g be real-valued FinSequence such that
    A1: for x be Nat holds f.x >= g.x & ex i be Nat st f.(i+1) > g.(i+1);
    consider i be Nat such that
    A2: f.(i+1) > g.(i+1) by A1;
    A5: Sum f = Sum (f|i) + f.(i+1) + Sum (f/^(i+1)) by SUM;
    A6: Sum g = Sum (g|i) + g.(i+1) + Sum (g/^(i+1)) by SUM;
    for x be Nat holds (f|i).x >= (g|i).x & (f/^(i+1)).x >= (g/^(i+1)).x
    proof
      let x be Nat;
      B0: i >= x implies (f|i).x = f.x & (g|i).x = g.x by FINSEQ_3:112;
      len (f|i) <= i & len (g|i) <= i by FINSEQ_5:17; then
      i < x implies len (f|i) < x & len (g|i) < x by XXREAL_0:2; then
      i < x implies not x in dom (f|i) & not x in dom (g|i)
        by FINSEQ_3:25; then
      B2: i < x implies (f|i).x = 0 & 0 = (g|i).x by FUNCT_1:def 2;
      x <> 0 implies (f.((i+1)+x) = (f/^(i+1)).x &
        g.((i+1)+x) = (g/^(i+1)).x) by FINSEQ74;
      hence thesis by A1,B0,B2;
    end; then
    Sum (f|i) >= Sum (g|i) & Sum(f/^(i+1)) >= Sum (g/^(i+1)) by NYS; then
    Sum (f|i) + Sum (f/^(i+1)) >= Sum (g|i) + Sum (g/^(i+1))
      by XREAL_1:7; then
    A7: Sum (f|i) + Sum (f/^(i+1)) + f.(i+1) >=
      Sum (g|i) + Sum (g/^(i+1))+f.(i+1) by XREAL_1:6;
    Sum (g|i) + Sum (g/^(i+1))+ f.(i+1) >
      Sum (g|i) + Sum (g/^(i+1))+ g.(i+1) by A2,XREAL_1:6;
    hence thesis by A5,A6,A7,XXREAL_0:2;
  end;
