reserve X for set;

theorem
for f being PartFunc of X,ExtREAL holds
 less_eq_dom(f,-infty) = eq_dom(f,-infty)
proof
   let f be PartFunc of X,ExtREAL;
   now let x be object;
    assume x in less_eq_dom(f,-infty); then
A1: x in dom f & -infty >= f.x by MESFUNC1:def 12; then
    f.x = -infty by XXREAL_0:6;
    hence x in eq_dom(f,-infty) by A1,MESFUNC1:def 15;
   end; then
A2:less_eq_dom(f,-infty) c= eq_dom(f,-infty);
   now let x be object;
    assume x in eq_dom(f,-infty); then
    x in dom f & -infty = f.x by MESFUNC1:def 15;
    hence x in less_eq_dom(f,-infty) by MESFUNC1:def 12;
   end; then
   eq_dom(f,-infty) c= less_eq_dom(f,-infty);
   hence less_eq_dom(f,-infty) = eq_dom(f,-infty) by A2;
end;
