reserve
  a for natural Number,
  k,l,m,n,k1,b,c,i for Nat,
  x,y,z,y1,y2 for object,
  X,Y for set,
  f,g for Function;
reserve p,q,r,s,t for FinSequence;
reserve D for set;

theorem Th38:
 for x being object holds p=<*x*> iff dom p = Seg 1 & rng p = {x}
proof let x be object;
  thus p = <*x*> implies dom p = Seg 1 & rng p = {x}
  proof
    assume
A1: p = <*x*>;
    hence dom p = Seg 1 by Def8;
    rng p = {p.1} by A1,Def8,Th2,FUNCT_1:4;
    hence thesis by A1;
  end;
  assume that
A2: dom p = Seg 1 and
A3: rng p = {x};
  1 in dom p by A2;
  then p.1 in {x} by A3,FUNCT_1:def 3;
  then p.1 = x by TARSKI:def 1;
  hence thesis by A2,Def8;
end;
