reserve x,y,z for set;
reserve f,f1,f2,f3 for FinSequence,
  p,p1,p2,p3 for set,
  i,k for Nat;
reserve D for non empty set,
  p,p1,p2,p3 for Element of D,
  f,f1,f2 for FinSequence of D;
reserve D for non empty set;
reserve p, q for FinSequence,
  X, Y, x, y for set,
  D for non empty set,
  i, j, k, l, m, n, r for Nat;

theorem Th13:
  q<>{} implies len (p^'q) +1 = len p + len q
proof
  set r = (p^'q);
  set qc = (2,len q)-cut q;
  assume q<>{};
  then 0+1<=len q by NAT_1:13;
  then 1+1<=len q +1 by XREAL_1:7;
  then
A1: len qc +(1+1) = len q + 1 by Lm2;
  thus len r +1 = len p + len qc + 1 by FINSEQ_1:22
    .= len p + len q by A1;
end;
