reserve p for Point of TOP-REAL 2,
  f,f1,f2,g for FinSequence of TOP-REAL 2,
  v, v1,v2 for FinSequence of REAL,
  r,s for Real,
  n,m,i,j,k for Nat,
  x for set;
reserve G for Go-board;

theorem
  x=G*(m,k) & x=G*(i,j) & [m,k] in Indices G & [i,j] in Indices G
  implies m=i & k=j
proof
  assume that
A1: x=G*(m,k) and
A2: x=G*(i,j) and
A3: [m,k] in Indices G and
A4: [i,j] in Indices G;
A5: len Line(G,m)=width G & dom Line(G,m)=Seg len Line(G,m) by FINSEQ_1:def 3
,MATRIX_0:def 7;
A6: Indices G=[:dom G,Seg width G:] by MATRIX_0:def 4;
  then
A7: k in Seg width G by A3,ZFMISC_1:87;
  then x=Line(G,m).k by A1,MATRIX_0:def 7;
  then
A8: x in rng Line(G,m) by A7,A5,FUNCT_1:def 3;
A9: len Col(G,k)=len G & dom Col(G,k)=Seg len Col(G,k) by FINSEQ_1:def 3
,MATRIX_0:def 8;
A10: len Line(G,i)=width G & dom Line(G,i)=Seg len Line(G,i) by FINSEQ_1:def 3
,MATRIX_0:def 7;
A11: len Col(G,j)=len G & dom Col(G,j)=Seg len Col(G,j) by FINSEQ_1:def 3
,MATRIX_0:def 8;
A12: dom G = Seg len G by FINSEQ_1:def 3;
A13: j in Seg width G by A4,A6,ZFMISC_1:87;
  then x=Line(G,i).j by A2,MATRIX_0:def 7;
  then
A14: x in rng Line(G,i) by A13,A10,FUNCT_1:def 3;
A15: i in dom G by A4,A6,ZFMISC_1:87;
  then x=Col(G,j).i by A2,MATRIX_0:def 8;
  then
A16: x in rng Col(G,j) by A15,A12,A11,FUNCT_1:def 3;
A17: m in dom G by A3,A6,ZFMISC_1:87;
  then x=Col(G,k).m by A1,MATRIX_0:def 8;
  then x in rng Col(G,k) by A17,A12,A9,FUNCT_1:def 3;
  hence thesis by A17,A15,A7,A13,A8,A14,A16,Th2,Th3;
end;
