
theorem Th1:
  for X be non empty set, f be nonpositive PartFunc of X,ExtREAL, E be set
    holds f|E is nonpositive
proof
  let X be non empty set, f be nonpositive PartFunc of X,ExtREAL, E be set;
  now let x be set;
    assume x in dom(f|E); then
    (f|E).x = f.x by FUNCT_1:47;
    hence (f|E).x <= 0 by MESFUNC5:8;
  end;
  hence f|E is nonpositive by MESFUNC5:9;
end;
