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 Th6:
  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 G2*(i2+1,j2)`1 <= G1*(i1+1,j1)`1
proof
  let G1,G2 be Go-board such that
A1: Values G1 c= Values G2 and
A2: 1 <= i1 and
A3: i1 < len G1 and
A4: 1 <= j1 & j1 <= width G1 and
A5: 1 <= i2 and
A6: i2 < len G2 and
A7: 1 <= j2 & j2 <= width G2 and
A8: G1*(i1,j1) = G2*(i2,j2);
  set p = G1*(i1+1,j1);
A9: i1+1 <= len G1 by A3,NAT_1:13;
  1 <= i1+1 by A2,NAT_1:13;
  then [i1+1,j1] in Indices G1 by A4,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
A10: p = G2*(i,j) and
A11: [i,j] in Indices G2;
A12: 1 <= i by A11,MATRIX_0:32;
A13: i <= len G2 by A11,MATRIX_0:32;
  1 <= j & j <= width G2 by A11,MATRIX_0:32;
  then
A14: G2*(i,j)`1 = G2*(i,1)`1 by A12,A13,GOBOARD5:2
    .= G2*(i,j2)`1 by A7,A12,A13,GOBOARD5:2;
  i1 < i1+1 by NAT_1:13;
  then
A15: G2*(i2,j2)`1 < G2*(i,j2)`1 by A2,A4,A8,A9,A10,A14,GOBOARD5:3;
A16: now
    assume i <= i2;
    then i = i2 or i < i2 by XXREAL_0:1;
    hence contradiction by A6,A7,A12,A15,GOBOARD5:3;
  end;
  assume
A17: p`1 < G2*(i2+1,j2)`1;
A18: 1 <= i2+1 by A5,NAT_1:13;
  now
    assume i2+1 <= i;
    then i2+1 = i or i2+1 < i by XXREAL_0:1;
    hence contradiction by A7,A17,A10,A13,A14,A18,GOBOARD5:3;
  end;
  hence contradiction by A16,NAT_1:13;
end;
