reserve f for non empty FinSequence of TOP-REAL 2,
  i,j,k,k1,k2,n,i1,i2,j1,j2 for Nat,
  r,s,r1,r2 for Real,
  p,q,p1,q1 for Point of TOP-REAL 2,
  G for Go-board;

theorem Th10:
  LSeg(f,k) is horizontal implies ex j st 1 <= j & j <= width GoB
  f & for p st p in LSeg(f,k) holds p`2 = (GoB f)*(1,j)`2
proof
  assume
A1: LSeg(f,k) is horizontal;
  per cases;
  suppose
A2: 1 <= k & k+1 <= len f;
    k <= k+1 by NAT_1:11;
    then k <= len f by A2,XXREAL_0:2;
    then k in dom f by A2,FINSEQ_3:25;
    then consider i,j such that
A3: [i,j] in Indices GoB f and
A4: f/.k = (GoB f)*(i,j) by GOBOARD2:14;
    take j;
    thus
A5: 1 <= j & j <= width GoB f by A3,MATRIX_0:32;
A6: f/.k in LSeg(f,k) by A2,TOPREAL1:21;
    let p;
A7: 1 <= i & i <= len GoB f by A3,MATRIX_0:32;
    assume p in LSeg(f,k);
    hence p`2 = (f/.k)`2 by A1,A6,SPPOL_1:def 2
      .= (GoB f)*(1,j)`2 by A4,A5,A7,GOBOARD5:1;
  end;
  suppose
A8: not(1 <= k & k+1 <= len f);
    take 1;
    width GoB f <> 0 by MATRIX_0:def 10;
    hence 1 <= 1 & 1 <= width GoB f by NAT_1:14;
    thus thesis by A8,TOPREAL1:def 3;
  end;
end;
