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 Th1:
  G is Y_equal-in-column & 1 <= j & j <= width G & 1 <= i & i <= len G
  implies G*(i,j)`2 = G*(1,j)`2
proof
  assume that
A1: G is Y_equal-in-column and
A2: 1 <= j and
A3: j <= width G and
A4: 1 <= i and
A5: i <= len G;
  j in Seg width G by A2,A3,FINSEQ_1:1;
  then
A6: Y_axis(Col(G,j)) is constant by A1;
  reconsider c = Col(G,j) as FinSequence of TOP-REAL 2;
A7: i in dom G by A4,A5,FINSEQ_3:25;
A8: 1 <= len G by A4,A5,XXREAL_0:2;
  then
A9: 1 in dom G by FINSEQ_3:25;
A10: len c = len G by MATRIX_0:def 8;
  then 1 in dom c by A8,FINSEQ_3:25;
  then
A11: c/.1 = c.1 by PARTFUN1:def 6;
  i in dom c by A4,A5,A10,FINSEQ_3:25;
  then
A12: c/.i = c.i by PARTFUN1:def 6;
A13: len(Y_axis Col(G,j)) = len c by GOBOARD1:def 2;
  then
A14: 1 in dom(Y_axis Col(G,j)) by A8,A10,FINSEQ_3:25;
A15: i in dom(Y_axis Col(G,j)) by A4,A5,A10,A13,FINSEQ_3:25;
  thus G*(i,j)`2 = (c/.i)`2 by A7,A12,MATRIX_0:def 8
    .= (Y_axis Col(G,j)).i by A15,GOBOARD1:def 2
    .= (Y_axis Col(G,j)).1 by A6,A14,A15
    .= (c/.1)`2 by A14,GOBOARD1:def 2
    .= G*(1,j)`2 by A9,A11,MATRIX_0:def 8;
end;
