
theorem Th37:
for f be PartFunc of REAL,REAL, a,b,c be Real st a < b <= c & ].a,c.] c= dom f
 & f is_left_improper_integrable_on a,c holds
  f is_left_improper_integrable_on a,b &
  ( left_improper_integral(f,a,c) = ext_left_integral(f,a,c)
     implies left_improper_integral(f,a,b) = ext_left_integral(f,a,b) ) &
  ( left_improper_integral(f,a,c) = +infty
     implies left_improper_integral(f,a,b) = +infty ) &
  ( left_improper_integral(f,a,c) = -infty
     implies left_improper_integral(f,a,b) = -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 is_left_improper_integrable_on a,c;

    per cases;
    suppose f is_left_ext_Riemann_integrable_on a,c; then
     left_improper_integral(f,a,c) = ext_left_integral(f,a,c) by A3,Th34;
     hence thesis by A1,A2,A3,Lm8;
    end;
    suppose not f is_left_ext_Riemann_integrable_on a,c; then
     per cases by A3,Th34;
     suppose left_improper_integral(f,a,c) = +infty;
      hence thesis by A1,A2,A3,Lm9;
     end;
     suppose left_improper_integral(f,a,c) = -infty;
      hence thesis by A1,A2,A3,Lm10;
     end;
    end;
end;
