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

theorem Th4:
  for f being FinSequence, x being set holds x in rng f or x..f = 0
proof
  let f be FinSequence, x be set;
  assume
A1: not x in rng f;
A2: now
    assume f " {x} <> {};
    then consider y being object such that
A3: y in f " {x} by XBOOLE_0:def 1;
    f.y in {x} by A3,FUNCT_1:def 7;
    then
A4: f.y = x by TARSKI:def 1;
    y in dom f by A3,FUNCT_1:def 7;
    hence contradiction by A1,A4,FUNCT_1:3;
  end;
  thus x..f = Sgm(f " {x}).1 by FINSEQ_4:def 4
    .= 0 by A2,FINSEQ_3:43;
end;
