
theorem Th48:
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 b be Real st a < b < c holds
  f is_left_improper_integrable_on a,b &
  f is_right_improper_integrable_on b,c &
  improper_integral(f,a,c)
   = left_improper_integral(f,a,b) + right_improper_integral(f,b,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 b be Real;
    assume
A3:  a < b < c;

    consider b1 be Real such that
A4:  a < b1 < c and
A5:  f is_left_improper_integrable_on a,b1 and
A6:  f is_right_improper_integrable_on b1,c and
A7:  improper_integral(f,a,c)
      = left_improper_integral(f,a,b1) + right_improper_integral(f,b1,c)
        by A1,A2,Def6;

    per cases;
    suppose A8: b < b1;
     ].a,b1.] c= ].a,c.[ by A4,XXREAL_1:49; then
A9:  ].a,b1.] c= dom f by A1;
     [.b,c.[ c= ].a,c.[ by A3,XXREAL_1:48; then
A10: [.b,c.[ c= dom f by A1;
A11: f|['b,b1'] is bounded & f is_integrable_on ['b,b1'] by A5,A3,A8;

     thus f is_left_improper_integrable_on a,b by A3,A5,A8,A9,Th37;
     thus f is_right_improper_integrable_on b,c by A4,A8,A6,A10,A11,Th43;
     thus thesis by A1,A2,A3,A4,A7,Th47;
    end;
    suppose A12: b >= b1;
     ].a,b.] c= ].a,c.[ by A3,XXREAL_1:49; then
A13:  ].a,b.] c= dom f by A1;
A14: f|['b1,b'] is bounded & f is_integrable_on ['b1,b'] by A6,A3,A12;
     [.b1,c.[ c= ].a,c.[ by A4,XXREAL_1:48; then
A15: [.b1,c.[ c= dom f by A1;

     thus f is_left_improper_integrable_on a,b by A4,A5,A12,A13,A14,Th38;
     thus f is_right_improper_integrable_on b,c by A12,A3,A15,A6,Th42;
     thus thesis by A1,A2,A3,A4,A7,Th47;
    end;
end;
