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 Th7:
  for G1,G2 being Go-board st G1*(i1-'1,j1) 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 G1*(i1-'1,j1)`1 <= G2*(i2-'1,j2)
  `1
proof
  let G1,G2 be Go-board such that
A1: G1*(i1-'1,j1) in Values G2 and
A2: 1 < i1 and
A3: i1 <= len G1 & 1 <= j1 & j1 <= width G1 and
A4: 1 < i2 and
A5: i2 <= len G2 and
A6: 1 <= j2 & j2 <= width G2 and
A7: G1*(i1,j1) = G2*(i2,j2);
  set p = G1*(i1-'1,j1);
A8: p in {G2*(i,j): [i,j] in Indices G2} by A1,MATRIX_0:39;
  1 <= i2-'1 by A4,NAT_D:49;
  then i2-'1 < i2 by NAT_D:51;
  then
A9: i2-'1 < len G2 by A5,XXREAL_0:2;
  consider i,j such that
A10: p = G2*(i,j) and
A11: [i,j] in Indices G2 by A8;
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 A6,A12,A13,GOBOARD5:2;
A15: 1 <= i1-'1 by A2,NAT_D:49;
  then i1-'1 < i1 by NAT_D:51;
  then
A16: G2*(i,j2)`1 < G2*(i2,j2)`1 by A3,A7,A15,A10,A14,GOBOARD5:3;
A17: now
    assume i2 <= i;
    then i = i2 or i2 < i by XXREAL_0:1;
    hence contradiction by A4,A6,A13,A16,GOBOARD5:3;
  end;
  assume
A18: G2*(i2-'1,j2)`1 < p`1;
  now
    assume i <= i2-'1;
    then i2-'1 = i or i < i2-'1 by XXREAL_0:1;
    hence contradiction by A6,A18,A10,A12,A14,A9,GOBOARD5:3;
  end;
  hence contradiction by A17,NAT_D:49;
end;
