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

theorem Th29:
  for f,g being non empty FinSequence of TOP-REAL 2 st 1 <= j & j+
  1 < len g holds LSeg(f^'g,len f+j) = LSeg(g,j+1)
proof
  let f,g be non empty FinSequence of TOP-REAL 2 such that
A1: 1 <= j and
A2: j+1 < len g;
A3: (f^'g)/.(len f +j+1) = (f^'g)/.(len f +(j+1))
    .= g/.(j+1+1) by A2,FINSEQ_6:160,NAT_1:11;
  j+0 <= j+1 by XREAL_1:6;
  then j < len g by A2,XXREAL_0:2;
  then
A4: (f^'g)/.(len f +j) = g/.(j+1) by A1,FINSEQ_6:160;
A5: 1 <= j+1 by NAT_1:11;
  len f + (j+1) < len f + len g by A2,XREAL_1:6;
  then len f +j+1 < len (f^'g)+1 by FINSEQ_6:139;
  then
A6: len f +j+1 <= len (f^'g) by NAT_1:13;
A7: j+1+1 <= len g by A2,NAT_1:13;
  j <= len f + j by NAT_1:11;
  then 1 <= len f +j by A1,XXREAL_0:2;
  hence LSeg(f^'g,len f+j) = LSeg((f^'g)/.(len f+j),(f^'g)/.(len f+j+1)) by A6,
TOPREAL1:def 3
    .= LSeg(g,j+1) by A4,A5,A3,A7,TOPREAL1:def 3;
end;
