theorem
  eq_dom(f,r) = f"{r}
proof
  now
    let x be object;
    assume
A1: x in f"{r};
    then f.x in {r} by FUNCT_1:def 7;
    then
A2: (R_EAL f).x = r by TARSKI:def 1;
    x in dom f by A1,FUNCT_1:def 7;
    hence x in eq_dom(f,r) by A2,MESFUNC1:def 15;
  end;
  then
A3: f"{r} c= eq_dom(f,r);
  now
    let x be object;
    assume
A4: x in eq_dom(f,r);
    then r = f.x by MESFUNC1:def 15;
    then
A5: f.x in {r} by TARSKI:def 1;
    x in dom f by A4,MESFUNC1:def 15;
    hence x in f"{r} by A5,FUNCT_1:def 7;
  end;
  then eq_dom(f,r) c= f"{r};
  hence thesis by A3;
end;
