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;

theorem Th36:
  len r = len p + len q & (for k being Nat st k in dom p holds r.k = p.k) &
  (for k being Nat st k in dom q holds r.(len p +  k) = q.k)
   implies r = p ^ q
proof
  assume len r = len p + len q;
  then
A1: dom r = Seg(len p + len q) by FINSEQ_1:def 3;
  assume that
A2: for k being Nat st k in dom p holds r.k = p.k and
A3: for k being Nat st k in dom q holds r.(len p + k) = q.k;
A4: for k being Nat st k in dom q holds r.(len p + k) = q.k by A3;
  for k being Nat st k in dom p holds r.k = p.k by A2;
  hence thesis by A1,A4,FINSEQ_1:def 7;
end;
