reserve p for Point of TOP-REAL 2,
  f,f1,f2,g for FinSequence of TOP-REAL 2,
  v, v1,v2 for FinSequence of REAL,
  r,s for Real,
  n,m,i,j,k for Nat,
  x for set;
reserve G for Go-board;
reserve D for set,
  f for FinSequence of D,
  M for Matrix of D;
reserve f for FinSequence of TOP-REAL 2;

theorem Th19:
  f is_sequence_on G & i in dom f implies ex n st n in dom G & f/.
  i in rng Line(G,n)
proof
  assume f is_sequence_on G & i in dom f;
  then consider n,m such that
A1: [n,m] in Indices G and
A2: f/.i=G*(n,m);
  set L = Line(G,n);
  take n;
A3: Indices G=[:dom G,Seg width G:] by MATRIX_0:def 4;
  hence n in dom G by A1,ZFMISC_1:87;
A4: m in Seg width G by A1,A3,ZFMISC_1:87;
  len L = width G by MATRIX_0:def 7;
  then
A5: m in dom L by A4,FINSEQ_1:def 3;
  L.m = f/.i by A2,A4,MATRIX_0:def 7;
  hence thesis by A5,FUNCT_1:def 3;
end;
