reserve x,y,y1,y2,z,a,b for object, X,Y,Z,V1,V2 for set,
  f,g,h,h9,f1,f2 for Function,
  i for Nat,
  P for Permutation of X,
  D,D1,D2,D3 for non empty set,
  d1 for Element of D1,
  d2 for Element of D2,
  d3 for Element of D3;

theorem Th15:
  f in PFuncs(X,PFuncs(Y,Z)) implies uncurry f in PFuncs([:X,Y:],Z
  ) & uncurry' f in PFuncs([:Y,X:],Z)
proof
  assume f in PFuncs(X,PFuncs(Y,Z));
  then
A1: ex g st f = g & dom g c= X & rng g c= PFuncs(Y,Z) by PARTFUN1:def 3;
  then dom uncurry f c= [:dom f,Y:] & [:dom f,Y:] c= [:X,Y:] by FUNCT_5:37
,ZFMISC_1:96;
  then
A2: dom uncurry f c= [:X,Y:];
  dom uncurry' f c= [:Y,dom f:] & [:Y,dom f:] c= [:Y,X:] by A1,FUNCT_5:37
,ZFMISC_1:96;
  then
A3: dom uncurry' f c= [:Y,X:];
  rng uncurry f c= Z & rng uncurry' f c= Z by A1,FUNCT_5:40;
  hence thesis by A2,A3,PARTFUN1:def 3;
end;
