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 Th52:
  i in dom D & f|A is bounded_below & g|A is bounded_below implies
  lower_volume(f,D).i + lower_volume(g,D).i <= lower_volume(f+g,D).i
proof
  assume that
A1: i in dom D and
A2: f|A is bounded_below and
A3: g|A is bounded_below;
A4: 0 <= vol(divset(D,i)) by SEQ_4:11,XREAL_1:48;
  dom(f+g) = A /\ A by FUNCT_2:def 1;
  then dom((f+g)|divset(D,i)) = divset(D,i) by A1,Th6,RELAT_1:62;
  then
A5: rng((f+g)|divset(D,i)) is non empty by RELAT_1:42;
  rng g is bounded_below by A3,Th9;
  then
A6: rng(g|divset(D,i)) is bounded_below by RELAT_1:70,XXREAL_2:44;
  dom g = A by FUNCT_2:def 1;
  then dom (g|divset(D,i)) = divset(D,i) by A1,Th6,RELAT_1:62;
  then
A7: rng(g|divset(D,i)) is non empty by RELAT_1:42;
  (f+g)|divset(D,i)=f|divset(D,i) + g|divset(D,i) by RFUNCT_1:44;
  then
A8: rng((f+g)|divset(D,i))c=rng(f|divset(D,i))++rng(g|divset(D,i)) by Th8;
  rng f is bounded_below by A2,Th9;
  then
A9: rng(f|divset(D,i)) is bounded_below by RELAT_1:70,XXREAL_2:44;
  then
A10: rng (f|divset(D,i))++rng(g|divset(D,i)) is bounded_below by A6,SEQ_4:124;
  dom f = A by FUNCT_2:def 1;
  then dom (f|divset(D,i)) = divset(D,i) by A1,Th6,RELAT_1:62;
  then rng(f|divset(D,i)) is non empty by RELAT_1:42;
  then lower_bound(rng(f|divset(D,i))++rng(g|divset(D,i))) =
    lower_bound rng(f|divset(D,i)) + lower_bound rng(g|divset(D,i))
     by A9,A6,A7,SEQ_4:125; then
  lower_bound rng((f+g)|divset(D,i))*vol(divset(D,i)) >= (lower_bound rng
  (f|divset(D,i)) + lower_bound rng(g|divset(D,i)))*vol(divset(D,i)) by A10,A5
,A4,A8,SEQ_4:47,XREAL_1:64;
  then
  lower_volume(f+g,D).i >= lower_bound rng(f|divset(D,i))*vol(divset(D,i)
  )+ lower_bound rng(g|divset(D,i))*vol(divset(D,i)) by A1,Def6;
  then
  lower_volume(f+g,D).i >= lower_volume(f,D).i+ lower_bound rng(g|divset(
  D,i))*vol(divset(D,i)) by A1,Def6;
  hence thesis by A1,Def6;
end;
