reserve i,k,n,m for Element of NAT;
reserve a,b,r,r1,r2,s,x,x1,x2 for Real;
reserve A for non empty closed_interval Subset of REAL;
reserve X for set;

theorem Th3:
  for F1,F2 being FinSequence of REAL st len F1 = len F2 & (for i
  st i in dom F1 holds F1.i <= F2.i) holds Sum F1 <= Sum F2
proof
  let F1,F2 be FinSequence of REAL;
  assume that
A1: len F1 = len F2 and
A2: for i st i in dom F1 holds F1.i <= F2.i;
  reconsider R1=F1 as Element of (len F1)-tuples_on REAL by FINSEQ_2:92;
  reconsider R2=F2 as Element of (len F1)-tuples_on REAL by A1,FINSEQ_2:92;
  for i be Nat st i in Seg len F1 holds R1.i <= R2.i
  proof
    let i be Nat;
    assume i in Seg len F1;
    then i in dom F1 by FINSEQ_1:def 3;
    hence thesis by A2;
  end;
  hence thesis by RVSUM_1:82;
end;
