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

theorem Th8:
  for f,g being FinSequence for x being set st x in rng f holds x..
  f = x..(f^'g)
proof
  let f,g be FinSequence, x be set;
  assume
A1: x in rng f;
  then
A2: f.(x..f) = x by FINSEQ_4:19;
A3: x..f in dom f by A1,FINSEQ_4:20;
  then
A4: x..f <= len f by FINSEQ_3:25;
A5: for i being Nat st 1 <= i & i < x..f holds (f^'g).i <> x
  proof
    let i be Nat such that
A6: 1 <= i and
A7: i < x..f;
A8: i < len f by A4,A7,XXREAL_0:2;
    then
A9: i in dom f by A6,FINSEQ_3:25;
    (f^'g).i = f.i by A6,A8,FINSEQ_6:140;
    hence thesis by A7,A9,FINSEQ_4:24;
  end;
  len f <= len(f^'g) by Th7;
  then
A10: dom f c= dom(f^'g) by FINSEQ_3:30;
  1 <= x..f by A3,FINSEQ_3:25;
  then (f^'g).(x..f) = x by A2,A4,FINSEQ_6:140;
  hence thesis by A3,A10,A5,FINSEQ_6:2;
end;
