reserve f for non empty FinSequence of TOP-REAL 2,
  i,j,k,k1,k2,n,i1,i2,j1,j2 for Nat,
  r,s,r1,r2 for Real,
  p,q,p1,q1 for Point of TOP-REAL 2,
  G for Go-board;

theorem
  1 <= i & i+1 <= len G & 1 <= j & j+1 <= width G implies 1/2*(G*(i,j)+G
  *(i+1,j+1)) = 1/2*(G*(i,j+1)+G*(i+1,j))
proof
  assume that
A1: 1 <= i and
A2: i+1 <= len G and
A3: 1 <= j and
A4: j+1 <= width G;
A5: j < width G by A4,NAT_1:13;
A6: 1 <= j+1 by NAT_1:11;
A7: 1 <= i+1 by NAT_1:11;
  then
A8: G*(i+1,j)`1 = G*(i+1,1)`1 by A2,A3,A5,GOBOARD5:2
    .= G*(i+1,j+1)`1 by A2,A4,A7,A6,GOBOARD5:2;
A9: i < len G by A2,NAT_1:13;
  then
A10: G*(i,j)`1 = G*(i,1)`1 by A1,A3,A5,GOBOARD5:2
    .= G*(i,j+1)`1 by A1,A4,A9,A6,GOBOARD5:2;
A11: G*(i+1,j+1)`2 = G*(1,j+1)`2 by A2,A4,A7,A6,GOBOARD5:1
    .= G*(i,j+1)`2 by A1,A4,A9,A6,GOBOARD5:1;
A12: G*(i,j)`2 = G*(1,j)`2 by A1,A3,A9,A5,GOBOARD5:1
    .= G*(i+1,j)`2 by A2,A3,A7,A5,GOBOARD5:1;
A13: (1/2*(G*(i,j)+G*(i+1,j+1)))`2 = 1/2*(G*(i,j)+G*(i+1,j+1))`2 by TOPREAL3:4
    .= 1/2*(G*(i,j)`2+G*(i+1,j+1)`2) by TOPREAL3:2
    .= 1/2*(G*(i,j+1)+G*(i+1,j))`2 by A12,A11,TOPREAL3:2
    .= (1/2*(G*(i,j+1)+G*(i+1,j)))`2 by TOPREAL3:4;
  (1/2*(G*(i,j)+G*(i+1,j+1)))`1 = 1/2*(G*(i,j)+G*(i+1,j+1))`1 by TOPREAL3:4
    .= 1/2*(G*(i,j)`1+G*(i+1,j+1)`1) by TOPREAL3:2
    .= 1/2*(G*(i,j+1)+G*(i+1,j))`1 by A10,A8,TOPREAL3:2
    .= (1/2*(G*(i,j+1)+G*(i+1,j)))`1 by TOPREAL3:4;
  hence
  1/2*(G*(i,j)+G*(i+1,j+1)) = |[(1/2*(G*(i,j+1)+G*(i+1,j)))`1,(1/2*(G*(i,
  j+1)+G*(i+1,j)))`2]| by A13,EUCLID:53
    .= 1/2*(G*(i,j+1)+G*(i+1,j)) by EUCLID:53;
end;
