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 Th4:
  for G1,G2 being Go-board st Values G1 c= Values G2 & [i1,j1] in
  Indices G1 & 1 <= i2 & i2 <= len G2 & G1*(i1,j1) = G2*(i2,1) holds j1 = 1
proof
  let G1,G2 be Go-board such that
A1: Values G1 c= Values G2 and
A2: [i1,j1] in Indices G1 and
A3: 1 <= i2 & i2 <= len G2 and
A4: G1*(i1,j1) = G2*(i2,1);
  set p = G1*(i1,1);
A5: 1 <= i1 & i1 <= len G1 by A2,MATRIX_0:32;
  assume
A6: j1 <> 1;
  1 <= j1 by A2,MATRIX_0:32;
  then
A7: 1 < j1 by A6,XXREAL_0:1;
  j1 <= width G1 by A2,MATRIX_0:32;
  then
A8: p`2 < G1*(i1,j1)`2 by A5,A7,GOBOARD5:4;
  0 <> width G1 by MATRIX_0:def 10;
  then 1 <= width G1 by NAT_1:14;
  then [i1,1] in Indices G1 by A5,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
A9: p = G2*(i,j) and
A10: [i,j] in Indices G2;
A11: 1 <= i & i <= len G2 by A10,MATRIX_0:32;
  0 <> width G2 by MATRIX_0:def 10;
  then
A12: 1 <= width G2 by NAT_1:14;
  then
A13: G2*(i,1)`2 = G2*(1,1)`2 by A11,GOBOARD5:1
    .= G2*(i2,1)`2 by A3,A12,GOBOARD5:1;
A14: j <= width G2 by A10,MATRIX_0:32;
  1 <= j by A10,MATRIX_0:32;
  then 1 < j by A4,A8,A9,A13,XXREAL_0:1;
  hence contradiction by A4,A8,A9,A11,A14,A13,GOBOARD5:4;
end;
