reserve p,q,r for FinSequence;
reserve u,v,x,y,y1,y2,z for object, A,D,X,Y for set;
reserve i,j,k,l,m,n for Nat;
reserve J for Nat;
reserve n for Nat;
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
  dom <:<*h*>:> = dom h & for x st x in dom h holds <:<*h*>:>.x = <*h.x *>
proof
A1: 1 in Seg 1 by FINSEQ_1:2,TARSKI:def 1;
A2: rng <:<*h*>:> c= product rngs <*h*> & rngs <*h*> = <*rng h*>
 by Th130,FUNCT_6:29;
  thus
A3: dom <:<*h*>:> = meet doms <*h*> by FUNCT_6:29
    .= meet <*dom h*> by Th130
    .= dom h by Th133;
  let x;
  assume
A4: x in dom h;
  then <:<*h*>:>.x in rng <:<*h*>:> by A3,FUNCT_1:def 3;
  then consider g such that
A5: <:<*h*>:>.x = g and
A6: dom g = dom <*rng h*> and
  for y being object st y in dom <*rng h*> holds g.y in <*rng h*>.y
by A2,CARD_3:def 5;
A7: dom g = Seg 1 by A6,FINSEQ_1:38;
  dom <*h*> = Seg 1 & <*h*>.1 = h by FINSEQ_1:38;
  then
A8: (uncurry <*h*>).(1,x) = h.x by A4,A1,FUNCT_5:38;
  reconsider g as FinSequence by A7,FINSEQ_1:def 2;
A9: len g = 1 by A7,FINSEQ_1:def 3;
  g.1 = (uncurry <*h*>).(1,x) by A3,A4,A5,A7,A1,FUNCT_6:31;
  hence thesis by A5,A8,A9,FINSEQ_1:40;
end;
