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 Th26:
 for f being Function-yielding Function holds
  x in dom <:f:> & g = <:f:>.x implies dom g = dom f &
  for y st y in dom g holds [y,x] in dom uncurry f & g.y = (uncurry f).(y,x)
proof let f being Function-yielding Function;
A1: rng <:f:> c= product rngs f & dom rngs f = dom f by Def2,Th25;
  assume
A2: x in dom <:f:> & g = <:f:>.x;
  then g in rng <:f:> by FUNCT_1:def 3;
  hence dom g = dom f by A1,CARD_3:9;
  let y such that
A3: y in dom g;
A4: [x,y] in dom ((uncurry' f)|([:meet doms f, dom f:] qua set)) by A2,A3,
FUNCT_5:31;
  then [x,y] in dom (uncurry' f) /\ [:meet doms f, dom f:] by RELAT_1:61;
  then
A5: [x,y] in dom uncurry' f by XBOOLE_0:def 4;
  g.y = ((uncurry' f)|([:meet doms f, dom f:] qua set)).(x,y) by A2,A3,
FUNCT_5:31;
  then
A6: g.y = (uncurry' f).(x,y) by A4,FUNCT_1:47;
  ~(uncurry f) = uncurry' f by FUNCT_5:def 4;
  hence thesis by A6,A5,FUNCT_4:42,43;
end;
