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 Th27:
  x in dom q implies ex k st k=x & len p + k in dom(p^q)
proof
  assume
A1: x in dom q;
  then
A2: x in Seg len q by Def3;
  reconsider k=x as Element of NAT by A1;
  take k;
A3: 1 <= k by A2,Th1;
A4: k <= len q by A2,Th1;
  k <= len p + k by NAT_1:11;
  then
A5: 1 <= len p + k by A3,XXREAL_0:2;
  len p + k <= len p + len q by A4,XREAL_1:7;
  then len p + k in Seg(len p + len q) by A5;
  hence thesis by Def7;
end;
