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 Th3:
  G is X_increasing-in-column &
  1 <= j & j <= width G & 1 <= i1 & i1 < i2 & i2 <= len G
  implies G*(i1,j)`1 < G*(i2,j)`1
proof
  assume that
A1: G is X_increasing-in-column and
A2: 1 <= j and
A3: j <= width G and
A4: 1 <= i1 and
A5: i1 < i2 and
A6: i2 <= len G;
  j in Seg width G by A2,A3,FINSEQ_1:1;
  then
A7: X_axis(Col(G,j)) is increasing by A1;
  reconsider c = Col(G,j) as FinSequence of TOP-REAL 2;
A8: i1 <= len G by A5,A6,XXREAL_0:2;
  then
A9: i1 in dom G by A4,FINSEQ_3:25;
A10: 1 <= i2 by A4,A5,XXREAL_0:2;
  then
A11: i2 in dom G by A6,FINSEQ_3:25;
A12: len c = len G by MATRIX_0:def 8;
  then i1 in dom c by A4,A8,FINSEQ_3:25;
  then
A13: c/.i1 = c.i1 by PARTFUN1:def 6;
  i2 in dom c by A6,A10,A12,FINSEQ_3:25;
  then
A14: c/.i2 = c.i2 by PARTFUN1:def 6;
A15: len(X_axis Col(G,j)) = len c by GOBOARD1:def 1;
  then
A16: i1 in dom(X_axis Col(G,j)) by A4,A8,A12,FINSEQ_3:25;
A17: G*(i1,j)`1 = (c/.i1)`1 by A9,A13,MATRIX_0:def 8
    .= (X_axis Col(G,j)).i1 by A16,GOBOARD1:def 1;
A18: i2 in dom(X_axis Col(G,j)) by A6,A10,A12,A15,FINSEQ_3:25;
  then (X_axis Col(G,j)).i2 = (c/.i2)`1 by GOBOARD1:def 1
    .= G*(i2,j)`1 by A11,A14,MATRIX_0:def 8;
  hence thesis by A5,A7,A16,A17,A18;
end;
