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 Th16:
  1 <= i & i+1 <= len G & 1 <= j & j+1 <= width G implies G*(i,j)
`1 = G*(i,j+1)`1 & G*(i,j)`2 = G*(i+1,j)`2 & G*(i+1,j+1)`1 = G*(i+1,j)`1 & G*(i
  +1,j+1)`2 = G*(i,j+1)`2
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: i < len G by A2,NAT_1:13;
  hence G*(i,j)`1 = G*(i,1)`1 by A1,A3,A5,GOBOARD5:2
    .= G*(i,j+1)`1 by A1,A4,A7,A6,GOBOARD5:2;
A8: 1 <= i+1 by NAT_1:11;
  thus G*(i,j)`2 = G*(1,j)`2 by A1,A3,A7,A5,GOBOARD5:1
    .= G*(i+1,j)`2 by A2,A3,A5,A8,GOBOARD5:1;
  thus G*(i+1,j+1)`1 = G*(i+1,1)`1 by A2,A4,A8,A6,GOBOARD5:2
    .= G*(i+1,j)`1 by A2,A3,A5,A8,GOBOARD5:2;
  thus G*(i+1,j+1)`2 = G*(1,j+1)`2 by A2,A4,A8,A6,GOBOARD5:1
    .= G*(i,j+1)`2 by A1,A4,A7,A6,GOBOARD5:1;
end;
