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 Th44:
  C is connected implies GoB Cage(C,n) = Gauge(C,n)
proof
A1: S-max L~Cage(C,n) in rng Cage(C,n) & E-max L~Cage(C,n) in rng Cage(C,n)
  by SPRECT_2:42,46;
  assume
A2: C is connected;
  then consider iw being Nat such that
A3: 1 <= iw & iw <= width Gauge(C,n) and
A4: W-min L~Cage(C,n) = Gauge(C,n)*(1,iw) by JORDAN1D:30;
A5: N-min L~Cage(C,n) in rng Cage(C,n) & Cage(C,n) is_sequence_on Gauge(C,n
  ) by A2,JORDAN9:def 1,SPRECT_2:39;
  consider ie being Nat such that
A6: 1 <= ie & ie <= width Gauge(C,n) and
A7: Gauge(C,n)*(len Gauge(C,n),ie) = E-max L~Cage(C,n) by A2,JORDAN1D:25;
A8: 1 <= len Gauge(C,n) by GOBRD11:34;
  then
A9: [len Gauge(C,n),ie] in Indices Gauge(C,n) by A6,MATRIX_0:30;
  consider iS being Nat such that
A10: 1 <= iS & iS <= len Gauge(C,n) and
A11: Gauge(C,n)*(iS,1) = S-max L~Cage(C,n) by A2,JORDAN1D:28;
A12: 1 <= width Gauge(C,n) by GOBRD11:34;
  then
A13: [iS,1] in Indices Gauge(C,n) by A10,MATRIX_0:30;
  consider IN being Nat such that
A14: 1 <= IN & IN <= len Gauge(C,n) and
A15: Gauge(C,n)*(IN,width Gauge(C,n)) = N-min L~Cage(C,n) by A2,JORDAN1D:21;
A16: [IN,width Gauge(C,n)] in Indices Gauge(C,n) by A12,A14,MATRIX_0:30;
  [1,iw] in Indices Gauge(C,n) by A8,A3,MATRIX_0:30;
  hence thesis by A4,A11,A13,A7,A9,A1,A15,A16,A5,Th34,SPRECT_2:43;
end;
