theorem
  for a,b being object, f being Function holds a.-->b c= f iff a in dom f &
  f.a = b
proof
  let a,b be object, f be Function;
A1: dom(a.-->b) = {a};
A2: a in dom(a.-->b) by TARSKI:def 1;
  hereby
    assume
A3: a.-->b c= f;
    then {a} c= dom f by A1,GRFUNC_1:2;
    hence a in dom f by ZFMISC_1:31;
    thus f.a = (a.-->b).a by A2,A3,GRFUNC_1:2
      .= b by Th72;
  end;
  assume that
A4: a in dom f and
A5: f.a = b;
A6: now
    let x be object;
    assume x in dom(a.-->b);
    then x = a by TARSKI:def 1;
    hence (a.-->b).x = f.x by A5,Th72;
  end;
  dom(a.-->b) c= dom f by A4,ZFMISC_1:31;
  hence thesis by A6,GRFUNC_1:2;
end;
