reserve G for Go-board,
  i,j,k,m,n for Nat;

theorem Th15:
  for p being set, D being non empty set for f being non empty
FinSequence of D for g being FinSequence of D st p..f = len f holds (f^g)-|p =
  (1,len f -' 1)-cut f
proof
  let p be set, D be non empty set, f be non empty FinSequence of D, g be
  FinSequence of D such that
A1: p..f = len f;
  p in rng f by A1,Th4;
  then
A2: p..(f^g) = p..f by FINSEQ_6:6;
  reconsider i = len f - 1 as Element of NAT by INT_1:5,NAT_1:14;
A3: len f -' 1 = i by NAT_1:14,XREAL_1:233;
  len(f^g) = len f + len g by FINSEQ_1:22;
  then
A4: len f <= len(f^g) by NAT_1:11;
  rng f c= rng(f^g) by FINSEQ_1:29;
  hence (f^g)-|p = (f^g)|Seg i by A1,A2,Th4,FINSEQ_4:33
    .= (f^g)|i by FINSEQ_1:def 16
    .= (1,len f -' 1)-cut(f^g) by A4,A3,Th14,NAT_D:44
    .= (1,len f -' 1)-cut f by MSSCYC_1:2,NAT_D:44;
end;
