reserve i,i1,i2,i9,i19,j,j1,j2,j9,j19,k,k1,k2,l,m,n for Nat;
reserve r,s,r9,s9 for Real;
reserve D for non empty set, f for FinSequence of D;
reserve f for FinSequence of TOP-REAL 2, G for Go-board;

theorem Th8:
  for G1,G2 being Go-board st G1*(i1,j1+1) in Values G2 & 1 <= i1
  & i1 <= len G1 & 1 <= j1 & j1 < width G1 & 1 <= i2 & i2 <= len G2 & 1 <= j2 &
j2 < width G2 & G1*(i1,j1) = G2*(i2,j2) holds G2*(i2,j2+1)`2 <= G1*(i1,j1+1)`2
proof
  let G1,G2 be Go-board such that
A1: G1*(i1,j1+1) in Values G2 and
A2: 1 <= i1 & i1 <= len G1 & 1 <= j1 and
A3: j1 < width G1 and
A4: 1 <= i2 & i2 <= len G2 and
A5: 1 <= j2 and
A6: j2 < width G2 and
A7: G1*(i1,j1) = G2*(i2,j2);
  set p = G1*(i1,j1+1);
  p in {G2*(i,j): [i,j] in Indices G2} by A1,MATRIX_0:39;
  then consider i,j such that
A8: p = G2*(i,j) and
A9: [i,j] in Indices G2;
A10: 1 <= j by A9,MATRIX_0:32;
A11: j <= width G2 by A9,MATRIX_0:32;
  1 <= i & i <= len G2 by A9,MATRIX_0:32;
  then
A12: G2*(i,j)`2 = G2*(1,j)`2 by A10,A11,GOBOARD5:1
    .= G2*(i2,j)`2 by A4,A10,A11,GOBOARD5:1;
  j1 < j1+1 & j1+1 <= width G1 by A3,NAT_1:13;
  then
A13: G2*(i2,j2)`2 < G2*(i2,j)`2 by A2,A7,A8,A12,GOBOARD5:4;
A14: now
    assume j <= j2;
    then j = j2 or j < j2 by XXREAL_0:1;
    hence contradiction by A4,A6,A10,A13,GOBOARD5:4;
  end;
  assume
A15: p`2 < G2*(i2,j2+1)`2;
A16: 1 <= j2+1 by A5,NAT_1:13;
  now
    assume j2+1 <= j;
    then j2+1 = j or j2+1 < j by XXREAL_0:1;
    hence contradiction by A4,A15,A8,A11,A12,A16,GOBOARD5:4;
  end;
  hence contradiction by A14,NAT_1:13;
end;
