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 Th16:
  1<len q implies (p^'q).len (p^'q) = q.len q
proof
  set r = p^'q;
  set qc = (2,len q)-cut q;
  assume
A1: 1<len q;
  then 1+1<=len q by NAT_1:13;
  then
A2: 1+1-1<=len q -1 by XREAL_1:9;
  q<>{} by A1;
  then len r +1-1=len p +len q -1 by Th13;
  then
A3: len r = len p +(len q -1);
  1+1<=len q +1 by A1,XREAL_1:7;
  then len qc +(1+1) = len q + 1 by Lm2;
  then
A4: len qc +1+1 = len q + 1;
  then len qc < len q by NAT_1:13;
  hence thesis by A3,A4,A2,Th15;
end;
