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 Th61:
  1 <= k & k <= len Cage(C,n) & 1 <= t & t <= width (Gauge(C,n)) &
Cage(C,n)/.k = Gauge(C,n)*(len Gauge(C,n),t) implies Cage(C,n)/.k in E-most L~
  Cage(C,n)
proof
  assume that
A1: 1 <= k & k <= len Cage(C,n) and
A2: 1 <= t & t <= width (Gauge(C,n)) and
A3: Cage(C,n)/.k = Gauge(C,n)*(len Gauge(C,n),t);
  Cage(C,n) is_sequence_on Gauge(C,n) by JORDAN9:def 1;
  then
A4: (Gauge(C,n)*(len Gauge(C,n),t))`1 >= E-bound L~Cage(C,n) by A2,Th23;
  len Cage(C,n) >= 2 by GOBOARD7:34,XXREAL_0:2;
  then
A5: Cage(C,n)/.k in L~Cage(C,n) by A1,TOPREAL3:39;
  then E-bound L~Cage(C,n) >= (Cage(C,n)/.k)`1 by PSCOMP_1:24;
  hence thesis by A3,A5,A4,SPRECT_2:13,XXREAL_0:1;
end;
