reserve x,y for Real,
  u,v,w for set,
  r for positive Real;

theorem
  for b being set, f being Function holds b in dom commute f iff ex a
  being set, g being Function st a in dom f & g = f.a & b in dom g
proof
  let b be set;
  let f be Function;
A1: dom commute f = proj2 dom uncurry f by FUNCT_5:23;
  hereby
    assume b in dom commute f;
    then consider a being object such that
A2: [a,b] in dom uncurry f by A1,XTUPLE_0:def 13;
    consider a9 being object, g being Function, b9 being object such that
A3: [a,b] = [a9,b9] and
A4: a9 in dom f and
A5: g = f.a9 and
A6: b9 in dom g by A2,FUNCT_5:def 2;
     reconsider a as set by TARSKI:1;
    take a, g;
    thus a in dom f & g = f.a & b in dom g by A3,A4,A5,A6,XTUPLE_0:1;
  end;
  given a being set, g being Function such that
A7: a in dom f and
A8: g = f.a and
A9: b in dom g;
  [a,b] in dom uncurry f by A7,A8,A9,FUNCT_5:def 2;
  hence thesis by A1,XTUPLE_0:def 13;
end;
