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 Th20:
  a<=b & f is_integrable_on [' a,b '] & f|[' a,b '] is bounded &
  [' a,b '] c= dom f & c in [' a,b '] & d in [' a,b '] implies integral(f,a,d)=
  integral(f,a,c)+integral(f,c,d)
proof
  assume
A1: a<=b & f is_integrable_on [' a,b '] & f|[' a,b '] is bounded & [' a,
  b '] c= dom f & c in [' a,b '] & d in [' a,b '];
  now
    assume
A2: not c <= d;
    then integral(f,a,c)=integral(f,a,d)+integral(f,d,c) by A1,Lm4;
    then
A3: integral(f,a,d) =integral(f,a,c) - integral(f,d,c);
    integral(f,c,d) = -integral(f,[' d,c ']) by A2,INTEGRA5:def 4;
    hence thesis by A2,A3,INTEGRA5:def 4;
  end;
  hence thesis by A1,Lm4;
end;
