
theorem Th47:
for f be PartFunc of REAL,REAL, a,c be Real st
 ].a,c.[ c= dom f & f is_improper_integrable_on a,c holds
for b1,b2 be Real st a < b1 < c & a < b2 < c holds
 left_improper_integral(f,a,b1)+right_improper_integral(f,b1,c)
  = left_improper_integral(f,a,b2)+right_improper_integral(f,b2,c)
proof
    let f be PartFunc of REAL,REAL, a,c be Real;
    assume that
A1:  ].a,c.[ c= dom f and
A2:  f is_improper_integrable_on a,c;
    let b1,b2 be Real;
    assume that
A3:  a < b1 < c and
A4:  a < b2 < c;

    consider b be Real such that
A5:  a < b < c and
A6:  f is_left_improper_integrable_on a,b and
A7:  f is_right_improper_integrable_on b,c and
A8:  not (left_improper_integral(f,a,b) = -infty
       & right_improper_integral(f,b,c) = +infty) and
A9:  not (left_improper_integral(f,a,b) = +infty
       & right_improper_integral(f,b,c) = -infty) by A2;

    per cases;
    suppose b1 < b & b2 < b; then
     left_improper_integral(f,a,b)+right_improper_integral(f,b,c)
      = left_improper_integral(f,a,b1)+right_improper_integral(f,b1,c)
   & left_improper_integral(f,a,b)+right_improper_integral(f,b,c)
      = left_improper_integral(f,a,b2)+right_improper_integral(f,b2,c)
       by A1,A3,A4,A5,A6,A7,A8,A9,Th45;
     hence thesis;
    end;
    suppose b1 < b & b <= b2; then
     left_improper_integral(f,a,b)+right_improper_integral(f,b,c)
      = left_improper_integral(f,a,b1)+right_improper_integral(f,b1,c)
   & left_improper_integral(f,a,b)+right_improper_integral(f,b,c)
      = left_improper_integral(f,a,b2)+right_improper_integral(f,b2,c)
       by A1,A3,A4,A5,A6,A7,A8,A9,Th45,Th46;
     hence thesis;
    end;
    suppose b <= b1 & b2 < b; then
     left_improper_integral(f,a,b)+right_improper_integral(f,b,c)
      = left_improper_integral(f,a,b1)+right_improper_integral(f,b1,c)
   & left_improper_integral(f,a,b)+right_improper_integral(f,b,c)
      = left_improper_integral(f,a,b2)+right_improper_integral(f,b2,c)
       by A1,A3,A4,A5,A6,A7,A8,A9,Th45,Th46;
     hence thesis;
    end;
    suppose b <= b1 & b <= b2; then
     left_improper_integral(f,a,b)+right_improper_integral(f,b,c)
      = left_improper_integral(f,a,b1)+right_improper_integral(f,b1,c)
   & left_improper_integral(f,a,b)+right_improper_integral(f,b,c)
      = left_improper_integral(f,a,b2)+right_improper_integral(f,b2,c)
       by A1,A3,A4,A5,A6,A7,A8,A9,Th46;
     hence thesis;
    end;
end;
