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 Th11:
  1 <= i & i < len G & 1 <= j & j < width G implies G*(i,j)+G*(i+1
  ,j+1) = G*(i,j+1)+G*(i+1,j)
proof
  assume that
A1: 1 <= i & i < len G and
A2: 1 <= j & j < width G;
A3: 1 <= j+1 & j+1 <= width G by A2,NAT_1:13;
A4: 1 <= i+1 & i+1 <= len G by A1,NAT_1:13;
  then
A5: G*(i+1,j+1)`1 = G*(i+1,1)`1 by A3,GOBOARD5:2
    .= G*(i+1,j)`1 by A2,A4,GOBOARD5:2;
A6: G*(i+1,j+1)`2 = G*(1,j+1)`2 by A4,A3,GOBOARD5:1
    .= G*(i,j+1)`2 by A1,A3,GOBOARD5:1;
A7: G*(i,j)`2 = G*(1,j)`2 by A1,A2,GOBOARD5:1
    .= G*(i+1,j)`2 by A2,A4,GOBOARD5:1;
A8: (G*(i,j)+G*(i+1,j+1))`2 = G*(i,j)`2+G*(i+1,j+1)`2 by Lm1
    .= (G*(i,j+1)+G*(i+1,j))`2 by A7,A6,Lm1;
A9: G*(i,j)`1 = G*(i,1)`1 by A1,A2,GOBOARD5:2
    .= G*(i,j+1)`1 by A1,A3,GOBOARD5:2;
  (G*(i,j)+G*(i+1,j+1))`1 = G*(i,j)`1+G*(i+1,j+1)`1 by Lm1
    .= (G*(i,j+1)+G*(i+1,j))`1 by A9,A5,Lm1;
  hence
  G*(i,j)+G*(i+1,j+1) = |[(G*(i,j+1)+G*(i+1,j))`1,(G*(i,j+1)+G*(i+1,j))`2
  ]| by A8,EUCLID:53
    .= G*(i,j+1)+G*(i+1,j) by EUCLID:53;
end;
