reserve i,j,k,n for Nat,
  D for non empty set,
  f, g for FinSequence of D;
reserve G for Go-board,
  f, g for FinSequence of TOP-REAL 2,
  p for Point of TOP-REAL 2,
  r, s for Real,
  x for set;

theorem Th20:
  1 <= i & i+1 <= len G & 1 <= j & j+1 <= width G implies G*(i,j)
in cell(G,i,j) & G*(i,j+1) in cell(G,i,j) & G*(i+1,j+1) in cell(G,i,j) & G*(i+1
  ,j) in cell(G,i,j)
proof
  assume that
A1: 1 <= i and
A2: i+1 <= len G and
A3: 1 <= j and
A4: j+1 <= width G;
A5: i < i+1 & j < width G by A4,NAT_1:13;
  then
A6: G*(i,j)`1 <= G*(i+1,j)`1 by A1,A2,A3,GOBOARD5:3;
A7: G*(i,j)`1 <= G*(i+1,j)`1 by A1,A2,A3,A5,GOBOARD5:3;
A8: j < j+1 & i < len G by A2,NAT_1:13;
  then
A9: G*(i,j)`2 <= G*(i,j+1)`2 by A1,A3,A4,GOBOARD5:4;
A10: G*(i+1,j+1)`1 = G*(i+1,j)`1 by A1,A2,A3,A4,Th16;
  then
A11: G*(i,j)`1 <= G*(i+1,j+1)`1 by A1,A2,A3,A5,GOBOARD5:3;
  G*(i,j)`2 <= G*(i,j+1)`2 by A1,A3,A4,A8,GOBOARD5:4;
  hence G*(i,j) in cell(G,i,j) by A1,A2,A3,A4,A6,Th17;
A12: G*(i,j)`1 = G*(i,j+1)`1 by A1,A2,A3,A4,Th16;
  then G*(i,j+1)`1 <= G*(i+1,j)`1 by A1,A2,A3,A5,GOBOARD5:3;
  hence G*(i,j+1) in cell(G,i,j) by A1,A2,A3,A4,A12,A9,Th17;
A13: G*(i+1,j+1)`2 = G*(i,j+1)`2 by A1,A2,A3,A4,Th16;
  then G*(i,j)`2 <= G*(i+1,j+1)`2 by A1,A3,A4,A8,GOBOARD5:4;
  hence G*(i+1,j+1) in cell(G,i,j) by A1,A2,A3,A4,A10,A11,A13,Th17;
A14: G*(i,j)`2 = G*(i+1,j)`2 by A1,A2,A3,A4,Th16;
  then G*(i+1,j)`2 <= G*(i,j+1)`2 by A1,A3,A4,A8,GOBOARD5:4;
  hence thesis by A1,A2,A3,A4,A7,A14,Th17;
end;
