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 Th17:
  a <= b & f is_integrable_on [' a,b '] & f|[' a,b '] is bounded &
  [' a,b '] c= dom f & c in [' a,b '] implies f is_integrable_on ['a,c '] & f
  is_integrable_on ['c,b '] & integral(f,a,b )=integral(f,a,c )+integral(f,c,b)
proof
  assume that
A1: a <= b and
A2: f is_integrable_on [' a,b '] and
A3: f|[' a,b '] is bounded & [' a,b '] c= dom f & c in [' a,b '];
  now
    assume
A4: b=c;
    then
A5: [' c,b ']= [.c,b.] by INTEGRA5:def 3;
    thus f is_integrable_on ['a,c '] by A2,A4;
    [' c,b ']= [. lower_bound([' c,b ']),upper_bound([' c,b ']) .]
    by INTEGRA1:4;
    then lower_bound([' c,b ']) = c & upper_bound([' c,b ']) = b
    by A5,INTEGRA1:5;
    then
A6: vol(['c,b ']) =0 by A4;
    then f||['c,b '] is integrable by INTEGRA4:6;
    hence f is_integrable_on ['c,b '];
    integral(f,c,b ) = integral(f,[' c,b '] ) by A5,INTEGRA5:19;
    then integral(f,c,b ) =0 by A6,INTEGRA4:6;
    hence integral(f, a,b )= integral(f,a,c ) +integral(f,c,b) by A4;
  end;
  hence thesis by A1,A2,A3,Lm2;
end;
