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 & 1 <= j & j <= width Gauge(E,n) implies LSeg(Gauge(E,
n)*(Center Gauge(E,n),1), Gauge(E,n)*(Center Gauge(E,n),j)) c= LSeg(Gauge(E,m)*
  (Center Gauge(E,m),1), Gauge(E,n)*(Center Gauge(E,n),j))
proof
  set a = N-bound E, s = S-bound E, w = W-bound E, e = E-bound E, G = Gauge(E,
  n), M = Gauge(E,m), sn = Center G, sm = Center M;
  assume that
A1: 1 <= m and
A2: m <= n and
A3: 1 <= j and
A4: j <= width G;
  now
A5: 0 < a - s by SPRECT_1:32,XREAL_1:50;
    then
A6: s-(a-s)/(2|^m) <= s-0 by XREAL_1:13;
A7: (a-s)/(2|^n) <= (a-s)/(2|^m) by A2,A5,Lm7;
A8: 1 <= len M by GOBRD11:34;
    then [sm,1] in Indices M by Lm4;
    then
A9: M*(sm,1)`2 = s-(a-s)/(2|^m) by Lm11;
    let t be Nat;
    assume that
A10: 1 <= t and
A11: t <= j;
    1 <= sn & sn <= len G by Lm3;
    then
A12: G*(sn,t)`2 <= G*(sn,j)`2 by A4,A10,A11,SPRECT_3:12;
A13: len G = width G by JORDAN8:def 1;
    then
A14: t <= len G by A4,A11,XXREAL_0:2;
    then
A15: M*(sm,1)`1 = G*(sn,t)`1 by A1,A2,A10,A8,Th36;
A16: [sn,t] in Indices G by A10,A14,Lm4;
    then
A17: G*(sn,t)`2 = |[w+(e-w)/(2|^n)*(sn - 2), s+(a-s)/(2|^n)*(t - 2)]| `2
    by JORDAN8:def 1
      .= s+(a-s)/(2|^n)*(t-2) by EUCLID:52;
A18: now
      per cases by A10,XXREAL_0:1;
      suppose
        t = 1;
        then G*(sn,t)`2 = s-(a-s)/(2|^n) by A16,Lm11;
        hence M*(sm,1)`2 <= G*(sn,t)`2 by A7,A9,XREAL_1:13;
      end;
      suppose
        t > 1;
        then t >= 1+1 by NAT_1:13;
        then t-2 >= 2-2 by XREAL_1:9;
        then s+0 <= s+(a-s)/(2|^n)*(t-2) by A5,XREAL_1:6;
        hence M*(sm,1)`2 <= G*(sn,t)`2 by A17,A6,A9,XXREAL_0:2;
      end;
    end;
    G*(sn,t)`1 = G*(sn,j)`1 by A1,A2,A3,A4,A10,A13,A14,Th36;
    hence G*(sn,t) in LSeg(M*(sm,1),G*(sn,j)) by A15,A18,A12,GOBOARD7:7;
  end;
  then
  G*(sn,1) in LSeg(M*(sm,1),G*(sn,j)) & G*(sn,j) in LSeg(M*(sm,1),G*(sn,j
  )) by A3;
  hence thesis by TOPREAL1:6;
end;
