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
  A c= B implies vol(A) <= vol(B)
proof
  assume
A1: A c= B;
  vol(A) = upper_bound A - lower_bound A by INTEGRA1:def 5;
  then upper_bound B >= vol(A) + lower_bound A by A1,SEQ_4:48;
  then
A2: upper_bound B - vol(A) >= lower_bound A by XREAL_1:19;
  lower_bound A >= lower_bound B by A1,SEQ_4:47;
  then upper_bound B - vol(A) >= lower_bound B by A2,XXREAL_0:2;
  then upper_bound B - lower_bound B >= vol(A) by XREAL_1:11;
  hence thesis by INTEGRA1:def 5;
end;
