
theorem Th83:
  for R1, R2 being real-valued FinSequence st
  len R1 = len R2 &
  (for j be Nat st j in Seg len R1 holds R1.j <= R2.j) &
  (ex j be Nat st j in Seg len R1 & R1.j < R2.j) holds
    Sum R1 < Sum R2
  proof
    let R1, R2 be real-valued FinSequence;
    set i = len R1;
    assume
A1: len R1 = len R2 &
    (for j be Nat st j in Seg i holds R1.j <= R2.j) &
    (ex j be Nat st j in Seg i & R1.j < R2.j);
    reconsider w = len R1 as natural Number;
    R1 is FinSequence of REAL & R2 is FinSequence of REAL by RVSUM_1:145; then
    reconsider r1 = R1, r2 = R2 as Element of w-tuples_on REAL
      by A1,FINSEQ_2:92;
    Sum r1 < Sum r2 by A1,RVSUM_1:83;
    hence thesis;
  end;
