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 Th24:
  f is_sequence_on G & i in dom f & i+1 in dom f & n in Seg width
G & f/.i in rng Col(G,n) implies f/.(i+1) in rng Col(G,n) or for k st f/.(i+1)
  in rng Col(G,k) & k in Seg width 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 Seg width G & f/.i in rng Col(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;
A9: j2 in Seg width G by A5,A7,ZFMISC_1:87;
  len Col(G,j2) = len G by MATRIX_0:def 8;
  then
A10: j1 in dom Col(G,j2) by A8,FINSEQ_3:29;
  consider i1,i2 be Nat such that
A11: [i1,i2] in Indices G and
A12: f/.i=G*(i1,i2) by A1,A2;
A13: i1 in dom G by A11,A7,ZFMISC_1:87;
  len Col(G,i2) = len G by MATRIX_0:def 8;
  then
A14: i1 in dom Col(G,i2) by A13,FINSEQ_3:29;
  Col(G,i2).i1 = f/.i by A12,A13,MATRIX_0:def 8;
  then
A15: f/.i in rng Col(G,i2) by A14,FUNCT_1:def 3;
  i2 in Seg width G by A11,A7,ZFMISC_1:87;
  then i2=n by A4,A15,Th3;
  then
A16: |.i1-j1.|+|.n-j2.| = 1 by A1,A2,A3,A11,A12,A5,A6;
A17: Col(G,j2).j1=f/.(i+1) by A6,A8,MATRIX_0:def 8;
  then
A18: f/.(i+1) in rng Col(G,j2) by A10,FUNCT_1:def 3;
  now
    per cases by A16,SEQM_3:42;
    suppose
      |.i1-j1.|=1 & n=j2;
      hence thesis by A17,A10,FUNCT_1:def 3;
    end;
    suppose
      |.n-j2.|=1 & i1=j1;
      hence thesis by A9,A18,Th3;
    end;
  end;
  hence thesis;
end;
