reserve X,Y,Z,X1,X2,Y1,Y2 for set, x,y,z,t,x1,x2 for object,
  f,g,h,f1,f2,g1,g2 for Function;

theorem
  x in dom f & g = f.x & y in dom g implies [y,x] in dom uncurry' f & (
  uncurry' f).(y,x) = g.y & g.y in rng uncurry' f
proof
  assume
A1: x in dom f & g = f.x & y in dom g;
  then [x,y] in dom uncurry f by Th31;
  hence
A2: [y,x] in dom uncurry' f by FUNCT_4:42;
  (uncurry f).(x,y) = g.y by A1,Th31;
  hence (uncurry' f).(y,x) = g.y by A2,FUNCT_4:43;
  hence thesis by A2,FUNCT_1:def 3;
end;
