reserve X for non empty set,
  Y for set,
  S for SigmaField of X,
  F for sequence of S,
  f,g for PartFunc of X,REAL,
  A,B for Element of S,
  r,s for Real,
  a for Real,
  n for Nat;
reserve X for non empty set,
  S for SigmaField of X,
  f,g for PartFunc of X,REAL,
  A for Element of S,
  r for Real,
  p for Rational;
reserve X for non empty set,
  f,g for PartFunc of X,REAL,
  r for Real ;
reserve X for non empty set,
  S for SigmaField of X,
  f,g for PartFunc of X,REAL,
  A for Element of S;
reserve X for non empty set,
  Y for set,
  S for SigmaField of X,
  f,g,h for PartFunc of X,REAL,
  A for Element of S,
  r for Real;

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;
