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
  1 <= m & m <= n implies LSeg(Gauge(E,n)*(Center Gauge(E,n),1), Gauge(E
,n)*(Center Gauge(E,n),len Gauge(E,n))) c= LSeg(Gauge(E,m)*(Center Gauge(E,m),1
  ), Gauge(E,m)*(Center Gauge(E,m),len Gauge(E,m)))
proof
  set a = N-bound E, s = S-bound E, G = Gauge(E,n), M = Gauge(E,m), sn =
  Center G, sm = Center M;
  assume
A1: 1 <= m;
A2: 1 <= len M by GOBRD11:34;
  then [sm,1] in Indices M by Lm4;
  then
A3: M*(sm,1)`2 = s-(a-s)/(2|^m) by Lm11;
  [sm,len M] in Indices M by A2,Lm4;
  then
A4: M*(sm,len M)`2 = a+(a-s)/(2|^m) by Lm13;
A5: sn <= len G by Lm3;
A6: 1 <= len G by GOBRD11:34;
  assume
A7: m <= n;
  then
A8: M*(sm,1)`1 = G*(sn,len G)`1 & G*(sn,len G)`1 = M*(sm,len M)`1 by A1,A6,A2
,Th36;
  0 < a - s by SPRECT_1:32,XREAL_1:50;
  then
A9: (a-s)/(2|^n) <= (a-s)/(2|^m) by A7,Lm7;
  len G = width G & 1 <= sn by Lm3,JORDAN8:def 1;
  then
A10: G*(sn,1)`2 <= G*(sn,len G)`2 by A6,A5,SPRECT_3:12;
  [sn,len G] in Indices G by A6,Lm4;
  then G*(sn,len G)`2 = a+(a-s)/(2|^n) by Lm13;
  then
A11: G*(sn,len G)`2 <= M*(sm,len M)`2 by A9,A4,XREAL_1:7;
  then
A12: G*(sn,1)`2 <= M*(sm,len M)`2 by A10,XXREAL_0:2;
  [sn,1] in Indices G by A6,Lm4;
  then G*(sn,1)`2 = s-(a-s)/(2|^n) by Lm11;
  then
A13: M*(sm,1)`2 <= G*(sn,1)`2 by A9,A3,XREAL_1:13;
  then M*(sm,1)`2 <= G*(sn,len G)`2 by A10,XXREAL_0:2;
  then
A14: G*(sn,len G) in LSeg(M*(sm,1),M*(sm,len M)) by A11,A8,GOBOARD7:7;
  M*(sm,1)`1 = G*(sn,1)`1 & G*(sn,1)`1 = M*(sm,len M)`1 by A1,A7,A6,A2,Th36;
  then G*(sn,1) in LSeg(M*(sm,1),M*(sm,len M)) by A13,A12,GOBOARD7:7;
  hence thesis by A14,TOPREAL1:6;
end;
