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 Th29:
 for f being Function-yielding Function
  st x in dom f & g = f.x & y in dom <:f:> & h = <:f:>.y
 holds g.y = h.x
proof let f be Function-yielding Function;
  assume that
A1: x in dom f & g = f.x and
A2: y in dom <:f:> and
A3: h = <:f:>.y;
  dom h = dom f by A2,A3,Th26;
  then x in dom h by A1;
  then
A4: h.x = (uncurry f).(x,y) by A2,A3,Th26;
  g in rng f by A1,FUNCT_1:def 3;
  then y in dom g by A2,Th27;
  hence thesis by A1,A4,FUNCT_5:38;
end;
