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 Th18:
  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 & ['c,d '] c= dom f
proof
  assume that
A1: a <= c and
A2: c <= d and
A3: d <= b and
A4: f is_integrable_on [' a,b '] and
A5: f|[' a,b '] is bounded and
A6: [' a,b '] c= dom f;
  a <= d by A1,A2,XXREAL_0:2;
  then
A7: a <= b by A3,XXREAL_0:2;
A8: c <= b by A2,A3,XXREAL_0:2;
  then c in [' a,b '] by A1,Lm3;
  then
A9: f is_integrable_on ['c,b '] by A4,A5,A6,A7,Th17;
A10: d in [' c,b '] by A2,A3,Lm3;
A11: ['c,b '] c= [' a,b '] by A1,A8,Lm3;
  then ['c,b '] c= dom f & f|[' c,b '] is bounded by A5,A6,RFUNCT_1:74;
  hence f is_integrable_on ['c,d '] by A8,A10,A9,Th17;
  [' c,d '] c= [' c,b '] by A2,A3,Lm3;
  then
A12: [' c,d '] c= [' a,b '] by A11;
  hence f|[' c,d '] is bounded by A5,RFUNCT_1:74;
  thus thesis by A6,A12;
end;
