
theorem Th38:
for f be PartFunc of REAL,REAL, a,b,c be Real
 st a < b <= c & ].a,c.] c= dom f & f|['b,c'] is bounded
  & f is_left_improper_integrable_on a,b & f is_integrable_on ['b,c']
holds f is_left_improper_integrable_on a,c &
  ( left_improper_integral(f,a,b) = ext_left_integral(f,a,b) implies
     left_improper_integral(f,a,c)
     = left_improper_integral(f,a,b) + integral(f,b,c) ) &
  ( left_improper_integral(f,a,b) = +infty implies
     left_improper_integral(f,a,c) = +infty ) &
  ( left_improper_integral(f,a,b) = -infty implies
     left_improper_integral(f,a,c) = -infty )
proof
    let f be PartFunc of REAL,REAL, a,b,c be Real;
    assume that
A1:  a < b <= c and
A2:  ].a,c.] c= dom f and
A3:  f|['b,c'] is bounded and
A4:  f is_left_improper_integrable_on a,b and
A5:  f is_integrable_on ['b,c'];
    per cases;
    suppose f is_left_ext_Riemann_integrable_on a,b; then
     left_improper_integral(f,a,b) = ext_left_integral(f,a,b) by A4,Th34;
     hence thesis by A1,A2,A3,A4,A5,Lm12;
    end;
    suppose not f is_left_ext_Riemann_integrable_on a,b; then
     per cases by A4,Th34;
     suppose left_improper_integral(f,a,b) = +infty;
      hence thesis by A1,A2,A3,A4,A5,Lm13;
     end;
     suppose left_improper_integral(f,a,b) = -infty;
      hence thesis by A1,A2,A3,A4,A5,Lm14;
     end;
    end;
end;
