reserve i, i1, i2, j, j1, j2, k, m, n, t for Nat,
  D for non empty Subset of TOP-REAL 2,
  E for compact non vertical non horizontal Subset of TOP-REAL 2,
  C for compact connected non vertical non horizontal Subset of TOP-REAL 2,
  G for Go-board,
  p, q, x for Point of TOP-REAL 2,
  r, s for Real;

theorem Th55:
  ex k,t st 1 <= k & k < len Cage(C,n) & 1 <= t & t <= width (
  Gauge(C,n)) & Cage(C,n)/.k = Gauge(C,n)*(1,t)
proof
  consider k,t such that
A1: 1 <= k and
A2: k <= len Cage(C,n) and
A3: 1 <= t & t <= width (Gauge(C,n)) and
A4: Cage(C,n)/.k = Gauge(C,n)*(1,t) by Lm15;
  per cases by A2,XXREAL_0:1;
  suppose
    k<len Cage(C,n);
    hence thesis by A1,A3,A4;
  end;
  suppose
A5: k=len Cage(C,n);
    take 1,t;
    thus 1 <= 1 & 1 < len Cage(C,n) by GOBOARD7:34,XXREAL_0:2;
    thus 1 <= t & t <= width (Gauge(C,n)) by A3;
    thus thesis by A4,A5,FINSEQ_6:def 1;
  end;
end;
