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 Th18:
  k in dom (p^q) implies (k in dom p or ex n st n in dom q & k=len
  p + n )
proof
  assume k in dom(p^q);
  then k in Segm(len p + len q) by Def3;
  then
A1: k < len p + len q by NAT_1:44;
  now
    assume len p <= k;
    then consider n being Nat such that
A2: k=len p + n by NAT_1:10;
    n + len p - len p < len q + len p - len p by A1,A2,XREAL_1:14;
    hence thesis by A2,Lm1;
  end;
  hence thesis by Lm1;
end;
