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 Th27:
  k in Seg width G & f/.1 in rng Col(G,1) & f is_sequence_on G & (
  for i st i in dom f & f/.i in rng Col(G,k) holds n<=i) implies for i st i in
dom f & i<=n holds for m st m in Seg width G & f/.i in rng Col(G,m) holds m<=k
proof
  assume that
A1: k in Seg width G and
A2: f/.1 in rng Col(G,1) and
A3: f is_sequence_on G and
A4: for i st i in dom f & f/.i in rng Col(G,k) holds n<=i;
  defpred P[Nat] means
$1 in dom f & $1<=n implies for m st m in
  Seg width G & f/.$1 in rng Col(G,m) holds m<=k;
A5: dom G = Seg len G by FINSEQ_1:def 3;
  0<width G by MATRIX_0:44;
  then 0+1<=width G by NAT_1:13;
  then
A6: 1 in Seg width G by FINSEQ_1:1;
A7: 1<=k by A1,FINSEQ_1:1;
A8: for i st P[i] holds P[i+1]
  proof
    let i such that
A9: P[i];
    assume that
A10: i+1 in dom f and
A11: i+1<=n;
    let m such that
A12: m in Seg width G & f/.(i+1) in rng Col(G,m);
    now
      per cases;
      suppose
        i=0;
        hence thesis by A2,A6,A7,A12,Th3;
      end;
      suppose
A13:    i<>0;
        i+1<=len f by A10,FINSEQ_3:25;
        then
A14:    i<=len f by NAT_1:13;
A15:    i<n by A11,NAT_1:13;
A16:    0+1<=i by A13,NAT_1:13;
        then
A17:    i in dom f by A14,FINSEQ_3:25;
        then consider i1,i2 be Nat such that
A18:    [i1,i2] in Indices G and
A19:    f/.i = G*(i1,i2) by A3;
A20:    Indices G = [:dom G,Seg width G:] by MATRIX_0:def 4;
        then
A21:    i2 in Seg width G by A18,ZFMISC_1:87;
A22:    dom Col(G,i2) = Seg len Col(G,i2) & len Col(G,i2)=len G by
FINSEQ_1:def 3,MATRIX_0:def 8;
A23:    i1 in dom G by A18,A20,ZFMISC_1:87;
        then Col(G,i2).i1 = f/.i by A19,MATRIX_0:def 8;
        then
A24:    f/.i in rng Col(G,i2) by A5,A23,A22,FUNCT_1:def 3;
        then
A25:    i2<=k by A9,A11,A16,A14,A21,FINSEQ_3:25,NAT_1:13;
        now
          per cases by A25,XXREAL_0:1;
          case
A26:        i2<k;
            now
              per cases by A3,A10,A17,A21,A24,Th24;
              suppose
                f/.(i+1) in rng Col(G,i2);
                hence thesis by A12,A21,A26,Th3;
              end;
              suppose
                for j st f/.(i+1) in rng Col(G,j) & j in Seg width G
                holds |.i2-j.|=1;
                then
A27:            |.i2-m.|=1 by A12;
                now
                  per cases by A27,SEQM_3:41;
                  suppose
                    i2>m & i2=m+1;
                    hence thesis by A26,XXREAL_0:2;
                  end;
                  suppose
                    i2<m & m=i2+1;
                    hence thesis by A26,NAT_1:13;
                  end;
                end;
                hence thesis;
              end;
            end;
            hence thesis;
          end;
          case
            i2=k;
            hence contradiction by A4,A15,A17,A24;
          end;
        end;
        hence thesis;
      end;
    end;
    hence thesis;
  end;
A28: P[0] by FINSEQ_3:25;
  thus for n holds P[n] from NAT_1:sch 2(A28,A8);
end;
