reserve n for Nat,
  i,j for Nat,
  r,s,r1,s1,r2,s2,r9,s9 for Real,
  p,q for Point of TOP-REAL 2,
  G for Go-board,
  x,y for set,
  v for Point of Euclid 2;

theorem Th34:
  1 <= j & j+1 <= width G implies 1/2*(G*(len G,j)+G*(len G,j+1))+
  |[1,0]| in Int cell(G,len G,j)
proof
  assume that
A1: 1 <= j and
A2: j+1 <= width G;
  set s1 = G*(len G,j)`2, r1 = G*(len G,j)`1, s2 = G*(len G,j+1)`2;
  len G <> 0 by MATRIX_0:def 10;
  then
A3: 1 <= len G by NAT_1:14;
  j < width G by A2,NAT_1:13;
  then
A4: Int cell(G,len G,j) = { |[r,s]| : G*(len G,1)`1 < r & G*(1,j)`2 < s & s
  < G* (1,j+1)`2 } by A1,Th23;
  len G <> 0 by MATRIX_0:def 10;
  then
A5: 1 <= len G by NAT_1:14;
  j < j+1 by XREAL_1:29;
  then
A6: s1 < s2 by A1,A2,A5,GOBOARD5:4;
  then s1+s1 < s1+s2 by XREAL_1:6;
  then
A7: 1/2*(s1+s1) < 1/2*(s1+s2) by XREAL_1:68;
A8: j < width G by A2,NAT_1:13;
  then
A9: G*(len G,1)`1 = r1 by A1,A3,GOBOARD5:2;
  then
A10: G*(len G,1)`1 < r1+1 by XREAL_1:29;
A11: 1 <= j+1 by NAT_1:11;
  then G*(len G,1)`1 = G*(len G,j+1)`1 by A2,A3,GOBOARD5:2;
  then G*(len G,j) = |[r1,s1]| & G*(len G,j+1) = |[r1,s2]| by A9,EUCLID:53;
  then 1/2*(r1+r1) = r1 & G*(len G,j)+G*(len G,j+1) = |[r1+r1,s1+s2]| by
EUCLID:56;
  then 1/2*(G*(len G,j)+G*(len G,j+1))= |[r1,1/2*(s1+s2)]| by EUCLID:58;
  then
A12: 1/2*(G*(len G,j)+G*(len G,j+1))+|[1,0]| = |[r1+1,1/2*(s1+s2)+0]| by
EUCLID:56;
  s1+s2 < s2+s2 by A6,XREAL_1:6;
  then 1/2*(s1+s2) < 1/2*(s2+s2) by XREAL_1:68;
  then
A13: 1/2*(s1+s2) < G*(1,j+1)`2 by A2,A11,A3,GOBOARD5:1;
  G*(1,j)`2 = s1 by A1,A8,A3,GOBOARD5:1;
  hence thesis by A12,A7,A13,A10,A4;
end;
