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

theorem Th5:
  for a,b being set, f being Function holds a in dom ((commute f).b
  ) iff ex g being Function st a in dom f & g = f.a & b in dom g
proof
  let a,b be set;
  let f be Function;
  dom uncurry f c= [:dom f, proj1 union rng f:]
  proof
    let u be object;
    assume u in dom uncurry f;
    then consider a being object, g being Function, b being object such that
A1: u = [a,b] and
A2: a in dom f and
A3: g = f.a and
A4: b in dom g by FUNCT_5:def 2;
    g in rng f by A2,A3,FUNCT_1:def 3;
    then g c= union rng f by ZFMISC_1:74;
    then proj1 g c= proj1 union rng f by XTUPLE_0:8;
    hence thesis by A1,A2,A4,ZFMISC_1:def 2;
  end;
  then
A5: uncurry' commute f = uncurry f by FUNCT_5:50;
  hereby
    assume
A6: a in dom ((commute f).b);
    then b in dom commute f by FUNCT_1:def 2,RELAT_1:38;
    then [a,b] in dom uncurry f by A5,A6,FUNCT_5:39;
    then consider a9 being object, g being Function, b9 being object
    such that
A7: [a,b] = [a9,b9] and
A8: a9 in dom f and
A9: g = f.a9 and
A10: b9 in dom g by FUNCT_5:def 2;
    take g;
    thus a in dom f & g = f.a & b in dom g by A7,A8,A9,A10,XTUPLE_0:1;
  end;
  given g being Function such that
A11: a in dom f and
A12: g = f.a and
A13: b in dom g;
  [a,b] in dom uncurry f by A11,A12,A13,FUNCT_5:def 2;
  hence thesis by FUNCT_5:22;
end;
