reserve p,q for Point of TOP-REAL 2,
  i,i1,i2,j,j1,j2,k for Nat,
  r,s for Real,
  G for Matrix of TOP-REAL 2;

theorem Th2:
  G is X_equal-in-line & 1 <= j & j <= width G & 1 <= i & i <= len G
  implies G*(i,j)`1 = G*(i,1)`1
proof
  assume that
A1: G is X_equal-in-line and
A2: 1 <= j and
A3: j <= width G and
A4: 1 <= i and
A5: i <= len G;
  i in dom G by A4,A5,FINSEQ_3:25;
  then
A6: X_axis(Line(G,i)) is constant by A1;
  reconsider c = Line(G,i) as FinSequence of TOP-REAL 2;
A7: j in Seg width G by A2,A3,FINSEQ_1:1;
A8: 1 <= width G by A2,A3,XXREAL_0:2;
  then
A9: 1 in Seg width G by FINSEQ_1:1;
A10: len c = width G by MATRIX_0:def 7;
  then 1 in dom c by A8,FINSEQ_3:25;
  then
A11: c/.1 = c.1 by PARTFUN1:def 6;
  j in dom c by A2,A3,A10,FINSEQ_3:25;
  then
A12: c/.j = c.j by PARTFUN1:def 6;
A13: len(X_axis Line(G,i)) = len c by GOBOARD1:def 1;
  then
A14: 1 in dom(X_axis Line(G,i)) by A8,A10,FINSEQ_3:25;
A15: j in dom(X_axis Line(G,i)) by A2,A3,A10,A13,FINSEQ_3:25;
  thus G*(i,j)`1 = (c/.j)`1 by A7,A12,MATRIX_0:def 7
    .= (X_axis Line(G,i)).j by A15,GOBOARD1:def 1
    .= (X_axis Line(G,i)).1 by A6,A14,A15
    .= (c/.1)`1 by A14,GOBOARD1:def 1
    .= G*(i,1)`1 by A9,A11,MATRIX_0:def 7;
end;
