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 Th32:
  1 <= i & i+1 <= len G implies 1/2*(G*(i,width G)+G*(i+1,width G)
  )+|[0,1]| in Int cell(G,i,width G)
proof
  assume that
A1: 1 <= i and
A2: i+1 <= len G;
  set r1 = G*(i,width G)`1, s1 = G*(i,width G)`2, r2 = G*(i+1,width G)`1;
  width G <> 0 by MATRIX_0:def 10;
  then
A3: 1 <= width G by NAT_1:14;
  i < len G by A2,NAT_1:13;
  then
A4: Int cell(G,i,width G) = { |[r,s]| : G*(i,1)`1 < r & r < G*(i+1,1)`1 & G
  * (1,width G)`2 < s } by A1,Th25;
  width G <> 0 by MATRIX_0:def 10;
  then
A5: 1 <= width G by NAT_1:14;
  i < i+1 by XREAL_1:29;
  then
A6: r1 < r2 by A1,A2,A5,GOBOARD5:3;
  then r1+r1 < r1+r2 by XREAL_1:6;
  then
A7: 1/2*(r1+r1) < 1/2*(r1+r2) by XREAL_1:68;
A8: i < len G by A2,NAT_1:13;
  then
A9: G*(1,width G)`2 = s1 by A1,A3,GOBOARD5:1;
  then
A10: G*(1,width G)`2 < s1+1 by XREAL_1:29;
A11: 1 <= i+1 by NAT_1:11;
  then G*(1,width G)`2 = G*(i+1,width G)`2 by A2,A3,GOBOARD5:1;
  then G*(i,width G) = |[r1,s1]| & G*(i+1,width G) = |[r2,s1]| by A9,EUCLID:53;
  then 1/2*(s1+s1) = s1 & G*(i,width G)+G*(i+1,width G) = |[r1+r2,s1+s1]| by
EUCLID:56;
  then 1/2*(G*(i,width G)+G*(i+1,width G))= |[1/2*(r1+r2),s1]| by EUCLID:58;
  then
A12: 1/2*(G*(i,width G)+G*(i+1,width G))+|[0,1]| = |[1/2*(r1+r2)+0,s1+1 ]|
  by EUCLID:56;
  r1+r2 < r2+r2 by A6,XREAL_1:6;
  then 1/2*(r1+r2) < 1/2*(r2+r2) by XREAL_1:68;
  then
A13: 1/2*(r1+r2) < G*(i+1,1)`1 by A2,A11,A3,GOBOARD5:2;
  G*(i,1)`1 = r1 by A1,A8,A3,GOBOARD5:2;
  hence thesis by A12,A7,A13,A10,A4;
end;
