
theorem Th30:
  for X be non empty set, f be PartFunc of X,ExtREAL, a be R_eal
  holds eq_dom(f,a) = f"{a}
proof
  let X be non empty set;
  let f be PartFunc of X,ExtREAL;
  let a be R_eal;
  now
    let x be object;
    assume
A1: x in f"{a};
    then f.x in {a} by FUNCT_1:def 7;
    then
A2: f.x = a by TARSKI:def 1;
    x in dom f by A1,FUNCT_1:def 7;
    hence x in eq_dom(f,a) by A2,MESFUNC1:def 15;
  end;
  then
A3: f"{a} c= eq_dom(f,a);
  now
    let x be object;
    assume
A4: x in eq_dom(f,a);
    then f.x = a by MESFUNC1:def 15;
    then
A5: f.x in {a} by TARSKI:def 1;
    x in dom f by A4,MESFUNC1:def 15;
    hence x in f"{a} by A5,FUNCT_1:def 7;
  end;
  then eq_dom(f,a) c= f"{a};
  hence thesis by A3;
end;
