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
  C is connected implies N-min C in right_cell(Cage(C,n),1)
proof
  assume
A1: C is connected;
  then consider i such that
A2: 1 <= i and
A3: i+1 <= len Gauge(C,n) and
A4: Cage(C,n)/.1 = Gauge(C,n)*(i,width Gauge(C,n)) and
A5: Cage(C,n)/.2 = Gauge(C,n)*(i+1,width Gauge(C,n)) and
A6: N-min C in cell(Gauge(C,n),i,width Gauge(C,n)-'1) and
  N-min C <> Gauge(C,n)*(i,width Gauge(C,n)-'1) by JORDAN9:def 1;
A7: for i1,j1,i2,j2 being Nat st [i1,j1] in Indices GoB Cage(C,n)
  & [i2,j2] in Indices GoB Cage(C,n) & Cage(C,n)/.1 = (GoB Cage(C,n))*(i1,j1) &
  Cage(C,n)/.(1+1) = (GoB Cage(C,n))*(i2,j2) holds i1 = i2 & j1+1 = j2 & cell(
Gauge(C,n),i,width Gauge(C,n)-'1) = cell(GoB Cage(C,n),i1,j1) or i1+1 = i2 & j1
= j2 & cell(Gauge(C,n),i,width Gauge(C,n)-'1) = cell(GoB Cage(C,n),i1,j1-'1) or
i1 = i2+1 & j1 = j2 & cell(Gauge(C,n),i,width Gauge(C,n)-'1) = cell(GoB Cage(C,
  n),i2,j2) or i1 = i2 & j1 = j2+1 & cell(Gauge(C,n),i,width Gauge(C,n)-'1) =
  cell(GoB Cage(C,n),i1-'1,j2)
  proof
    0 <> width Gauge(C,n) by MATRIX_0:def 10;
    then
A8: 1 <= width Gauge(C,n) by NAT_1:14;
A9: GoB Cage(C,n) = Gauge(C,n) by A1,Th44;
    let i1,j1,i2,j2 be Nat such that
A10: [i1,j1] in Indices GoB Cage(C,n) and
A11: [i2,j2] in Indices GoB Cage(C,n) and
A12: Cage(C,n)/.1 = (GoB Cage(C,n))*(i1,j1) and
A13: Cage(C,n)/.(1+1) = (GoB Cage(C,n))*(i2,j2);
    1 <= i+1 by NAT_1:11;
    then
A14: [i+1,width Gauge(C,n)] in Indices Gauge(C,n) by A3,A8,MATRIX_0:30;
    then
A15: i2 = i+1 by A5,A11,A13,A9,GOBOARD1:5;
    i < len Gauge(C,n) by A3,NAT_1:13;
    then
A16: [i,width Gauge(C,n)] in Indices Gauge(C,n) by A2,A8,MATRIX_0:30;
    then
A17: i1 = i by A4,A10,A12,A9,GOBOARD1:5;
A18: j2 = width Gauge(C,n) by A5,A11,A13,A9,A14,GOBOARD1:5;
    per cases by A4,A10,A12,A9,A16,A15,A18,GOBOARD1:5;
    case
      i1 = i2 & j1+1 = j2;
      hence thesis by A17,A15;
    end;
    case
      i1+1 = i2 & j1 = j2;
      thus thesis by A4,A10,A12,A9,A16,A17,GOBOARD1:5;
    end;
    case
      i1 = i2+1 & j1 = j2;
      hence thesis by A17,A15;
    end;
    case
      i1 = i2 & j1 = j2+1;
      hence thesis by A17,A15;
    end;
  end;
  1+1 <= len Cage(C,n) by GOBOARD7:34,XXREAL_0:2;
  hence thesis by A6,A7,GOBOARD5:def 6;
end;
