reserve m,k,j,j1,i,i1,i2,n for Nat,
  r,s for Real,
  C for compact non vertical non horizontal Subset of TOP-REAL 2,
  G for Go-board,
  p for Point of TOP-REAL 2;

theorem
  for G be Go-board for f be FinSequence of TOP-REAL 2 st f
is_sequence_on G for k be Nat st 1 <= k & k+1 <= len f holds LSeg(f,
  k) c= left_cell(f,k,G)
proof
  let G be Go-board;
  let f be FinSequence of TOP-REAL 2;
  assume
A1: f is_sequence_on G;
  let k be Nat;
  assume
A2: 1 <= k & k+1 <= len f;
  then
A3: k in dom f by SEQ_4:134;
  then consider i1,j1 be Nat such that
A4: [i1,j1] in Indices G and
A5: f/.k = G*(i1,j1) by A1,GOBOARD1:def 9;
A6: k+1 in dom f by A2,SEQ_4:134;
  then consider i2,j2 be Nat such that
A7: [i2,j2] in Indices G and
A8: f/.(k+1) = G*(i2,j2) by A1,GOBOARD1:def 9;
A9: 1 <= i2 by A7,MATRIX_0:32;
A10: 1 <= j1 by A4,MATRIX_0:32;
  left_cell(f,k,G) = left_cell(f,k,G);
  then
A11: i1 = i2 & j1+1 = j2 & left_cell(f,k,G) = cell(G,i1-'1,j1) or i1+1 = i2
& j1 = j2 & left_cell(f,k,G) = cell(G,i1,j1) or i1 = i2+1 & j1 = j2 & left_cell
(f,k,G) = cell(G,i2,j2-'1) or i1 = i2 & j1 = j2+1 & left_cell(f,k,G) = cell(G,
  i1,j2) by A1,A2,A4,A5,A7,A8,GOBRD13:def 3;
A12: 1 <= j2 by A7,MATRIX_0:32;
A13: j1 <= width G by A4,MATRIX_0:32;
A14: j2 <= width G by A7,MATRIX_0:32;
A15: i2 <= len G by A7,MATRIX_0:32;
A16: 1 <= i1 by A4,MATRIX_0:32;
  |.i1-i2.|+|.j1-j2.| = 1 by A1,A3,A6,A4,A5,A7,A8,GOBOARD1:def 9;
  then
A17: |.i1-i2.|=1 & j1=j2 or |.j1-j2.|=1 & i1=i2 by SEQM_3:42;
A18: i1 <= len G by A4,MATRIX_0:32;
  per cases by A17,SEQM_3:41;
  suppose
A19: i1 = i2 & j1+1 = j2;
    then
A20: j1 < width G by A14,NAT_1:13;
A21: i1 -' 1 + 1 = i1 by A16,XREAL_1:235;
    then i1-'1 < len G by A18,NAT_1:13;
    then LSeg(f/.k,f/.(k+1)) c= cell(G,i1-'1,j1) by A5,A8,A10,A19,A21,A20,
GOBOARD5:18;
    hence thesis by A2,A11,A19,TOPREAL1:def 3;
  end;
  suppose
A22: i1+1 = i2 & j1 = j2;
    then i1 < len G by A15,NAT_1:13;
    then LSeg(f/.k,f/.(k+1)) c= cell(G,i1,j1) by A5,A8,A16,A10,A13,A22,
GOBOARD5:22;
    hence thesis by A2,A11,A22,TOPREAL1:def 3;
  end;
  suppose
A23: i1 = i2+1 & j1 = j2;
    then
A24: i2 < len G by A18,NAT_1:13;
A25: j2 -' 1 + 1 = j2 by A12,XREAL_1:235;
    then j2 -' 1 < width G by A14,NAT_1:13;
    then LSeg(f/.k,f/.(k+1)) c= cell(G,i2,j2-'1) by A5,A8,A9,A23,A25,A24,
GOBOARD5:21;
    hence thesis by A2,A11,A23,TOPREAL1:def 3;
  end;
  suppose
A26: i1 = i2 & j1 = j2+1;
    then j2 < width G by A13,NAT_1:13;
    then
    LSeg(f/.k,f/.(k+1)) c= left_cell(f,k,G) by A5,A8,A16,A18,A12,A11,A26,
GOBOARD5:19;
    hence thesis by A2,TOPREAL1:def 3;
  end;
end;
