reserve a,b,c,d,e,x,r for Real,
  A for non empty closed_interval Subset of REAL,
  f,g for PartFunc of REAL,REAL;

theorem
  a <= c & c <= d & d <= b & f is_integrable_on [' a,b '] & g
is_integrable_on [' a,b '] & f|[' a,b '] is bounded & g|[' a,b '] is bounded &
[' a,b '] c= dom f & [' a,b '] c= dom g implies f+g is_integrable_on ['c,d '] &
  (f+g)|[' c,d '] is bounded
proof
  assume that
A1: a <= c and
A2: c <= d & d <= b and
A3: f is_integrable_on [' a,b '] & g is_integrable_on [' a,b '] & f|[' a
  ,b '] is bounded & g|[' a,b '] is bounded and
A4: [' a,b '] c= dom f and
A5: [' a,b '] c= dom g;
A6: [' c,d '] c= [' c,b '] by A2,Lm3;
A7: f|[' c,d '] is bounded & g|[' c,d '] is bounded by A1,A2,A3,A4,A5,Th18;
  c <=b by A2,XXREAL_0:2;
  then
A8: ['c,b '] c= [' a,b '] by A1,Lm3;
  then ['c,b '] c= dom f by A4;
  then
A9: [' c,d '] c= dom f by A6;
  ['c,b '] c= dom g by A5,A8;
  then
A10: [' c,d '] c= dom g by A6;
  f is_integrable_on ['c,d '] & g is_integrable_on ['c,d '] by A1,A2,A3,A4,A5
,Th18;
  hence f+g is_integrable_on ['c,d '] by A7,A9,A10,Th11;
  (f+g)|([' c,d '] /\ ['c,d ']) is bounded by A7,RFUNCT_1:83;
  hence thesis;
end;
