reserve a,b,r,x,y for Real,
  i,j,k,n for Nat,
  x1 for set;
reserve A, B for non empty closed_interval Subset of REAL;
reserve f, g for Function of A,REAL;
reserve D, D1, D2 for Division of A;

theorem Th29:
  f|A is bounded_below & r >= 0 implies lower_sum(r(#)f,D) = r*
  lower_sum(f,D)
proof
  assume
A1: f|A is bounded_below & r >= 0;
A2: for i be Nat st 1 <= i & i <= len lower_volume(r(#)f,D) holds
  lower_volume(r(#)f,D).i = (r*lower_volume(f,D)).i
  proof
    let i be Nat;
    assume
A3: 1 <= i & i <= len lower_volume(r(#)f,D);
    len D = len lower_volume(r(#)f,D) by INTEGRA1:def 7;
    then i in dom D by A3,FINSEQ_3:25;
    then lower_volume(r(#)f,D).i = r*lower_volume(f,D).i by A1,Th25
      .= (r*lower_volume(f,D)).i by RVSUM_1:44;
    hence thesis;
  end;
  len lower_volume(r(#)f,D) = len D by INTEGRA1:def 7
    .= len lower_volume(f,D) by INTEGRA1:def 7
    .= len(r*lower_volume(f,D)) by NEWTON:2;
  then lower_volume(r(#)f,D)=r*lower_volume(f,D) by A2,FINSEQ_1:14;
  then lower_sum(r(#)f,D) =Sum(r*lower_volume(f,D)) by INTEGRA1:def 9
    .=r*Sum(lower_volume(f,D)) by RVSUM_1:87
    .=r*lower_sum(f,D) by INTEGRA1:def 9;
  hence thesis;
end;
