
theorem Th14:
  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_3:102; 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 MESFUNC5:11;
end;
