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 Th20:
  f is_sequence_on G & i in dom f & i+1 in dom f & n in dom G & f
/.i in rng Line(G,n) implies f/.(i+1) in rng Line(G,n) or for k st f/.(i+1) in
  rng Line(G,k) & k in dom G holds |.n-k.| = 1
proof
  assume that
A1: f is_sequence_on G and
A2: i in dom f and
A3: i+1 in dom f and
A4: n in dom G & f/.i in rng Line(G,n);
  consider j1,j2 be Nat such that
A5: [j1,j2] in Indices G and
A6: f/.(i+1)=G*(j1,j2) by A1,A3;
A7: Indices G=[:dom G,Seg width G:] by MATRIX_0:def 4;
  then
A8: j1 in dom G by A5,ZFMISC_1:87;
  consider i1,i2 be Nat such that
A9: [i1,i2] in Indices G and
A10: f/.i=G*(i1,i2) by A1,A2;
A11: i2 in Seg width G by A9,A7,ZFMISC_1:87;
  len Line(G,i1) = width G by MATRIX_0:def 7;
  then
A12: i2 in dom Line(G,i1) by A11,FINSEQ_1:def 3;
  Line(G,i1).i2 = f/.i by A10,A11,MATRIX_0:def 7;
  then
A13: f/.i in rng Line(G,i1) by A12,FUNCT_1:def 3;
  i1 in dom G by A9,A7,ZFMISC_1:87;
  then i1=n by A4,A13,Th2;
  then
A14: |.n-j1.|+|.i2-j2.| = 1 by A1,A2,A3,A9,A10,A5,A6;
A15: j2 in Seg width G by A5,A7,ZFMISC_1:87;
  len Line(G,j1) = width G by MATRIX_0:def 7;
  then
A16: j2 in dom Line(G,j1) by A15,FINSEQ_1:def 3;
A17: Line(G,j1).j2=f/.( i+1) by A6,A15,MATRIX_0:def 7;
  then
A18: f/.(i+1) in rng Line(G,j1) by A16,FUNCT_1:def 3;
  now
    per cases by A14,SEQM_3:42;
    suppose
      |.n-j1.|=1 & i2=j2;
      hence thesis by A8,A18,Th2;
    end;
    suppose
      |.i2-j2.|=1 & n=j1;
      hence thesis by A17,A16,FUNCT_1:def 3;
    end;
  end;
  hence thesis;
end;
