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
  X <> {} implies dom <:X --> f:> = dom f & for x st x in dom f holds <:
  X --> f:>.x = X --> f.x
proof
  assume
A1: X <> {};
  thus
A2: dom <:X --> f:> = meet doms (X --> f) by Th25
    .= meet (X --> dom f) by Th20
    .= dom f by A1,Th23;
A3: rng <:X --> f:> c= product rngs (X --> f) & rngs (X --> f) = X --> rng f
  by Th20,Th25;
  let x;
  assume
A4: x in dom f;
  then <:X --> f:>.x in rng <:X --> f:> by A2,FUNCT_1:def 3;
  then consider g such that
A5: <:X --> f:>.x = g and
A6: dom g = dom (X --> rng f) and
  for y being object st y in dom (X --> rng f) holds g.y in (X --> rng f).y
by A3,
CARD_3:def 5;
A7: dom g = X by A6;
A8: dom (X --> f) = X;
A9: now
    let y be object;
    assume
A10: y in X;
    then g.y = (uncurry (X --> f)).(y,x) & (X --> f).y = f by A2,A4,A5,A7,Th26,
FUNCOP_1:7;
    then g.y = f.x by A4,A8,A10,FUNCT_5:38;
    hence g.y = (X --> f.x).y by A10,FUNCOP_1:7;
  end;
  thus thesis by A5,A7,A9;
end;
