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

theorem Th6:
  for a,b being set, f,g being Function st a in dom f & g = f.a & b
  in dom g holds (commute f).b.a = g.b
proof
  let a,b be set;
  let f,g be Function;
  assume that
A1: a in dom f and
A2: g = f.a and
A3: b in dom g;
A4: [a,b] in dom uncurry f by A1,A2,A3,FUNCT_5:def 2;
A5: [a,b]`2 = b;
  [a,b]`1 = a;
  then (uncurry f).(a,b) = g.b by A5,A4,A2,FUNCT_5:def 2;
  hence thesis by A4,FUNCT_5:22;
end;
