reserve X for set;

theorem Th10:
for f being PartFunc of X,ExtREAL holds
 great_eq_dom(f,+infty) = eq_dom(f,+infty)
proof
   let f be PartFunc of X,ExtREAL;
   now let x be object;
    assume x in great_eq_dom(f,+infty); then
A1: x in dom f & +infty <= f.x by MESFUNC1:def 14; then
    f.x = +infty by XXREAL_0:4;
    hence x in eq_dom(f,+infty) by A1,MESFUNC1:def 15;
   end; then
A2:great_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 great_eq_dom(f,+infty) by MESFUNC1:def 14;
   end; then
   eq_dom(f,+infty) c= great_eq_dom(f,+infty);
   hence great_eq_dom(f,+infty) = eq_dom(f,+infty) by A2;
end;
