reserve X for set;
reserve n,i for Element of NAT;
reserve a,b,c,d,e,r,x0 for Real;
reserve A for non empty closed_interval Subset of REAL;
reserve f,g,h for PartFunc of REAL,REAL n;
reserve E for Element of REAL n;

theorem Th12:
  a <= c & c <= d & d <= b &
  f is_integrable_on ['a,b'] & f| ['a,b'] is bounded & ['a,b'] c= dom f
  implies
  -f is_integrable_on ['c,d'] & (-f) | ['c,d'] is bounded
  proof
    -f = (-1)(#)f by NFCONT_4:7;
    hence thesis by Th11;
  end;
