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 Th9:
  for G1,G2 being Go-board st Values G1 c= 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 G1*(i1,j1-'1)`2 <= G2*(i2,j2-'1)`2
proof
  let G1,G2 be Go-board such that
A1: Values G1 c= Values G2 and
A2: 1 <= i1 & i1 <= len G1 and
A3: 1 < j1 and
A4: j1 <= width G1 and
A5: 1 <= i2 & i2 <= len G2 and
A6: 1 < j2 and
A7: j2 <= width G2 and
A8: G1*(i1,j1) = G2*(i2,j2);
  set p = G1*(i1,j1-'1);
A9: 1 <= j1-'1 by A3,NAT_D:49;
  then
A10: j1-'1 < j1 by NAT_D:51;
  then j1-'1 < width G1 by A4,XXREAL_0:2;
  then [i1,j1-'1] in Indices G1 by A2,A9,MATRIX_0:30;
  then p in {G1*(i,j): [i,j] in Indices G1};
  then p in Values G1 by MATRIX_0:39;
  then p in Values G2 by A1;
  then p in {G2*(i,j): [i,j] in Indices G2} by MATRIX_0:39;
  then consider i,j such that
A11: p = G2*(i,j) and
A12: [i,j] in Indices G2;
A13: 1 <= j by A12,MATRIX_0:32;
A14: j <= width G2 by A12,MATRIX_0:32;
  1 <= i & i <= len G2 by A12,MATRIX_0:32;
  then
A15: G2*(i,j)`2 = G2*(1,j)`2 by A13,A14,GOBOARD5:1
    .= G2*(i2,j)`2 by A5,A13,A14,GOBOARD5:1;
  then
A16: G2*(i2,j)`2 < G2*(i2,j2)`2 by A2,A4,A8,A9,A10,A11,GOBOARD5:4;
A17: now
    assume j2 <= j;
    then j = j2 or j2 < j by XXREAL_0:1;
    hence contradiction by A5,A6,A14,A16,GOBOARD5:4;
  end;
  1 <= j2-'1 by A6,NAT_D:49;
  then j2-'1 < j2 by NAT_D:51;
  then
A18: j2-'1 < width G2 by A7,XXREAL_0:2;
  assume
A19: G2*(i2,j2-'1)`2 < p`2;
  now
    assume j <= j2-'1;
    then j2-'1 = j or j < j2-'1 by XXREAL_0:1;
    hence contradiction by A5,A19,A11,A13,A15,A18,GOBOARD5:4;
  end;
  hence contradiction by A17,NAT_D:49;
end;
