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,
m)*(Center Gauge(E,m),1), Gauge(E,n)*(Center Gauge(E,n),j)) 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, 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;
A5: 1 <= sm & sm <= len M by Lm3;
A6: 1 <= sn & sn <= len G by Lm3;
  then
A7: G*(sn,len G)`2 <= M*(sm,len M)`2 by A2,A5,Th40;
  len G = width G by JORDAN8:def 1;
  then G*(sn,j)`2 <= G*(sn,len G)`2 by A3,A4,A6,SPRECT_3:12;
  then
A8: G*(sn,j)`2 <= M*(sm,len M)`2 by A7,XXREAL_0:2;
A9: 0 < a - s by SPRECT_1:32,XREAL_1:50;
  then
A10: s-(a-s)/(2|^m) <= s-0 by XREAL_1:13;
A11: 1 <= len M by GOBRD11:34;
  then [sm,1] in Indices M by Lm4;
  then
A12: M*(sm,1)`2 = s-(a-s)/(2|^m) by Lm11;
A13: (a-s)/(2|^n) <= (a-s)/(2|^m) by A2,A9,Lm7;
A14: len G = width G by JORDAN8:def 1;
  then
A15: [sn,j] in Indices G by A3,A4,Lm4;
  then
A16: G*(sn,j)`2 = |[w+(e-w)/(2|^n)*(sn - 2), s+(a-s)/(2|^n)*(j - 2)]|`2 by
JORDAN8:def 1
    .= s+(a-s)/(2|^n)*(j-2) by EUCLID:52;
A17: now
    per cases by A3,XXREAL_0:1;
    suppose
      j = 1;
      then G*(sn,j)`2 = s-(a-s)/(2|^n) by A15,Lm11;
      hence M*(sm,1)`2 <= G*(sn,j)`2 by A13,A12,XREAL_1:13;
    end;
    suppose
      j > 1;
      then j >= 1+1 by NAT_1:13;
      then j-2 >= 2-2 by XREAL_1:9;
      then s+0 <= s+(a-s)/(2|^n)*(j-2) by A9,XREAL_1:6;
      hence M*(sm,1)`2 <= G*(sn,j)`2 by A12,A16,A10,XXREAL_0:2;
    end;
  end;
  len M = width M by JORDAN8:def 1;
  then
A18: M*(sm,1)`2 <= M*(sm,len M)`2 by A11,A5,SPRECT_3:12;
  M*(sm,1)`1 = M*(sm,len M)`1 by A1,A11,Th36;
  then
A19: M*(sm,1) in LSeg(M*(sm,1),M*(sm,len M)) by A18,GOBOARD7:7;
  M*(sm,1)`1 = G*(sn,j)`1 & G*(sn,j)`1 = M*(sm,len M)`1 by A1,A2,A3,A4,A11,A14
,Th36;
  then G*(sn,j) in LSeg(M*(sm,1),M*(sm,len M)) by A17,A8,GOBOARD7:7;
  hence thesis by A19,TOPREAL1:6;
end;
