reserve k,n for Nat,
  x,y,z,y1,y2 for object,X,Y for set,
  f,g for Function;
reserve p,q,r,s,t for XFinSequence;
reserve D for set;

theorem Th30:
  p=<%x%> iff dom p = Segm 1 & rng p = {x}
proof
  thus p = <%x%> implies dom p = Segm 1 & rng p = {x}
  proof
    assume
A1: p = <%x%>;
    hence dom p = Segm 1 by Def4;
    rng p = {p.0} by FUNCT_1:4,A1;
    hence thesis by A1,Def4;
  end;
  assume that
A2: dom p = Segm 1 and
A3: rng p = {x};
  1=0+1;
  then p.0 in {x} by A2,A3,FUNCT_1:3,NAT_1:45;
  then p.0 = x by TARSKI:def 1;
  hence thesis by A2,Def4;
end;
