
theorem Th13:
  for X be non empty set, r be Real, f be without-infty Function of X,ExtREAL
    st r >= 0 holds r(#)f is without-infty
proof
   let X be non empty set, r be Real, f be without-infty Function of X,ExtREAL;
   assume A1: r >= 0;
   now let x be set;
    assume A2: x in dom(r(#)f); then
A3: x in dom f by MESFUNC1:def 6;
    per cases by A1;
    suppose A4: r > 0; then
     r * f.x > r * -infty by A3,MESFUNC5:10,XXREAL_3:72; then
     r * f.x > -infty by A4,XXREAL_3:def 5;
     hence (r(#)f).x > -infty by A2,MESFUNC1:def 6;
    end;
    suppose r = 0; then
     r * f.x > -infty;
     hence (r(#)f).x > -infty by A2,MESFUNC1:def 6;
    end;
   end;
   hence r(#)f is without-infty by MESFUNC5:10;
end;
