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
  integral(exp_R,a,b) = exp_R.b-exp_R.a
proof
A1: min(a,b) <= a by XXREAL_0:17;
A2: [. min(a,b),max(a,b) .] c= REAL;
  exp_R|REAL is continuous & a <= max(a,b) by FDIFF_1:25,TAYLOR_1:16
,XXREAL_0:25;
  then
A3: exp_R.max(a,b) = integral(exp_R,min(a,b),max(a,b)) + exp_R.min(a,b) by A1
,A2,Th20,Th27,SIN_COS:47,XXREAL_0:2;
A4: min(a,b) = a implies exp_R.b - exp_R.a= integral(exp_R,a,b)
  proof
    assume
A5: min(a,b) = a;
    then max(a,b) = b by XXREAL_0:36;
    hence thesis by A3,A5;
  end;
  min(a,b) = b implies exp_R.b - exp_R.a= integral(exp_R,a,b)
  proof
    assume
A6: min(a,b) = b;
    then
A7: max(a,b) = a by XXREAL_0:36;
    b < a implies exp_R.b - exp_R.a= integral(exp_R,a,b)
    proof
      assume b < a;
      then integral(exp_R,a,b) = -integral(exp_R,[' b,a ']) by INTEGRA5:def 4;
      then exp_R.a = -integral(exp_R,a,b) + exp_R.b by A1,A3,A6,A7,
INTEGRA5:def 4;
      hence thesis;
    end;
    hence thesis by A1,A4,A6,XXREAL_0:1;
  end;
  hence thesis by A4,XXREAL_0:15;
end;
