reserve a,b,r for Real;
reserve A for non empty set;
reserve X,x for set;
reserve f,g,F,G for PartFunc of REAL,REAL;
reserve n for Element of NAT;

theorem Th20:
  a <= b & [. a,b .] c= X & X c= dom f & f|X is continuous &
    F is_integral_of f,X implies F.b = integral(f,a,b) + F.a
proof
  assume that
A1: a <= b and
A2: [. a,b .] c= X and
A3: X c= dom f;
  assume f|X is continuous;
  then
A4: f is_integrable_on [' a,b '] & f|[' a,b '] is bounded by A1,A2,A3,Th19;
A5: [. a,b .] =[' a,b '] by A1,INTEGRA5:def 3;
  assume F is_integral_of f,X;
  hence thesis by A1,A2,A5,A4,Th18;
end;
