reserve k for Element of NAT;
reserve r,r1 for Real;
reserve i for Integer;
reserve q for Rational;

theorem Th6:
  for C being non empty set, f being PartFunc of C,ExtREAL, r
  st r <> 0 holds
  for c being Element of C st c in dom(r(#)f) holds f.c = (r(#)f).c / r
proof
  let C being non empty set;
  let f being PartFunc of C,ExtREAL;
  let r;
  assume
A1: r <> 0;
  let c be Element of C;
  assume c in dom(r(#)f);
  then
A2: (r(#)f).c = r * f.c by Def6;
  per cases;
  suppose
A3: f.c = +infty;
 now per cases by A1;
      suppose
A4:     0. < r;
then      (r(#)f).c = +infty by A2,A3,XXREAL_3:def 5;
        hence thesis by A3,A4,XXREAL_3:83;
      end;
      suppose
A5:     r < 0.;
then     (r(#)f).c = -infty by A2,A3,XXREAL_3:def 5;
        hence thesis by A3,A5,XXREAL_3:84;
      end;
    end;
    hence thesis;
  end;
  suppose
A6: f.c = -infty;
 now per cases by A1;
      suppose
A7:    0. < r;
then     (r(#)f).c = -infty by A2,A6,XXREAL_3:def 5;
        hence thesis by A6,A7,XXREAL_3:86;
      end;
      suppose
A8:    r < 0.;
then     (r(#)f).c = +infty by A2,A6,XXREAL_3:def 5;
        hence thesis by A6,A8,XXREAL_3:85;
      end;
    end;
    hence thesis;
  end;
  suppose
 f.c <> +infty & f.c <> -infty;
    then reconsider a = f.c as Element of REAL by XXREAL_0:14;
    reconsider rr=r as R_eal by XXREAL_0:def 1;
    (r(#)f).c = (r qua ExtReal) * a by A2
       .= r * a;
    then  (r(#)f).c / rr = r*a/r by EXTREAL1:2
      .= a/(r/r) by XCMPLX_1:77
      .= a / 1 by A1,XCMPLX_1:60;
    hence thesis;
  end;
end;
