reserve a,a1,b,b1,x,y for Real,
  F,G,H for FinSequence of REAL,
  i,j,k,n,m for Element of NAT,
  I for Subset of REAL,
  X for non empty set,
  x1,R,s for set;
reserve A for non empty closed_interval Subset of REAL;
reserve A, B for non empty closed_interval Subset of REAL;
reserve r for Real;
reserve D, D1, D2 for Division of A;
reserve f, g for Function of A,REAL;

theorem Th54:
  f|A is bounded_below & g|A is bounded_below implies lower_sum(f,
  D) + lower_sum(g,D) <= lower_sum(f+g,D)
proof
  assume that
A1: f|A is bounded_below and
A2: g|A is bounded_below;
  set H=lower_volume(f+g,D);
  set G=lower_volume(g,D);
  set F=lower_volume(f,D);
  len G = len D by Def6;
  then
A3: G is Element of (len D)-tuples_on REAL by FINSEQ_2:92;
  len F = len D by Def6;
  then
A4: F is Element of (len D)-tuples_on REAL by FINSEQ_2:92;
A5: for j be Nat st j in Seg(len D) holds (F+G).j <= H.j
  proof
    let j be Nat;
    assume j in Seg(len D);
    then j in dom D by FINSEQ_1:def 3;
    then
    lower_volume(f,D).j + lower_volume(g,D).j <= lower_volume(f+g,D).j by A1,A2
,Th52;
    hence thesis by A4,A3,RVSUM_1:11;
  end;
  len H = len D by Def6;
  then
A6: H is Element of (len D)-tuples_on REAL by FINSEQ_2:92;
  F+G is Element of (len D)-tuples_on REAL by A4,A3,FINSEQ_2:120;
  then Sum(F+G) <= Sum H by A6,A5,RVSUM_1:82;
  hence thesis by A4,A3,RVSUM_1:89;
end;
