
theorem Th20:
  for X be non empty set, f be PartFunc of X,ExtREAL,c be Real st
f is nonnegative holds (0 <= c implies c(#)f is nonnegative) & (c <= 0 implies
  c(#)f is nonpositive)
proof
  let X be non empty set;
  let f be PartFunc of X,ExtREAL;
  let c be Real;
  set g = c(#)f;
  assume
A1: f is nonnegative;
  hereby
    set g = c(#)f;
    assume
A2: 0 <= c;
    for x be object st x in dom g holds 0 <= g.x
    proof
      let x be object;
      0 <= f.x by A1,SUPINF_2:51;
      then
A3:   0 <= c*f.x by A2;
      assume x in dom g;
      hence thesis by A3,MESFUNC1:def 6;
    end;
    hence c(#)f is nonnegative by SUPINF_2:52;
  end;
  assume
A4: c <= 0;
  now
    let x be set;
    0 <= f.x by A1,SUPINF_2:51;
    then
A5: c*f.x <= 0 by A4;
    assume x in dom g;
    hence g.x <= 0 by A5,MESFUNC1:def 6;
  end;
  hence thesis by Th9;
end;
