
theorem Th2:
for X be non empty set, A be set holds
  chi(+infty,A,X) = Xchi(A,X) & chi(-infty,A,X) = -Xchi(A,X)
proof
   let X be non empty set, A be set;
   for x be Element of X holds chi(+infty,A,X).x = Xchi(A,X).x
   proof
    let x be Element of X;
    per cases;
    suppose x in A; then
     chi(+infty,A,X).x = +infty & Xchi(A,X).x = +infty
       by Def1,MEASUR10:def 7;
     hence chi(+infty,A,X).x = Xchi(A,X).x;
    end;
    suppose not x in A; then
     chi(+infty,A,X).x = 0 & Xchi(A,X).x = 0 by Def1,MEASUR10:def 7;
     hence chi(+infty,A,X).x = Xchi(A,X).x;
    end;
   end;
   hence chi(+infty,A,X) = Xchi(A,X) by FUNCT_2:def 8;
   for x be Element of X holds chi(-infty,A,X).x = (-Xchi(A,X)).x
   proof
    let x be Element of X;
    x in X; then
A1: x in dom (-Xchi(A,X)) by FUNCT_2:def 1; then
A2: (-Xchi(A,X)).x = - (Xchi(A,X).x) by MESFUNC1:def 7;
    per cases;
    suppose x in A; then
     chi(-infty,A,X).x = -infty & Xchi(A,X).x = +infty
       by Def1,MEASUR10:def 7;
     hence chi(-infty,A,X).x = (-Xchi(A,X)).x by A1,XXREAL_3:6,MESFUNC1:def 7;
    end;
    suppose not x in A; then
     chi(-infty,A,X).x = 0 & Xchi(A,X).x = 0 by Def1,MEASUR10:def 7;
     hence chi(-infty,A,X).x = (-Xchi(A,X)).x by A2;
    end;
   end;
   hence chi(-infty,A,X) = -Xchi(A,X) by FUNCT_2:def 8;
end;
