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;
reserve f for standard special_circular_sequence;

theorem Th14:
  G is non empty-yielding X_equal-in-line X_increasing-in-column &
  1 <= i & i <= len G & 1 <= j & j < width G
  implies LSeg(G*(i,j),G*(i,j+1)) c= v_strip(G,i)
proof
  assume that
A1: G is non empty-yielding and
A2: G is X_equal-in-line and
A3: G is X_increasing-in-column and
A4: 1 <= i and
A5: i <= len G and
A6: 1 <= j and
A7: j < width G;
A8: 1 <= j+1 by A6,NAT_1:13;
A9: j+1 <= width G by A7,NAT_1:13;
  let x be object;
  assume
A10: x in LSeg(G*(i,j),G*(i,j+1));
  then reconsider p = x as Point of TOP-REAL 2;
A11: p = |[p`1, p`2]| by EUCLID:53;
A12: G*(i,j)`1 = G*(i,1)`1 by A2,A4,A5,A6,A7,Th2
    .= G*(i,j+1)`1 by A2,A4,A5,A8,A9,Th2;
  now per cases by A5,XXREAL_0:1;
    suppose
A13:  i = len G;
      then G*(len G,j)`1 <= p`1 by A10,A12,TOPREAL1:3;
      then p in { |[r,s]| : G*(len G,j)`1 <= r } by A11;
      hence thesis by A1,A2,A6,A7,A13,Th9;
    end;
    suppose
A14:  i < len G;
      then
A15:  i+1 <= len G by NAT_1:13;
A16:  G*(i,j)`1 <= p`1 by A10,A12,TOPREAL1:3;
      p`1 <= G*(i,j)`1 by A10,A12,TOPREAL1:3;
      then
A17:  p`1 = G*(i,j)`1 by A16,XXREAL_0:1;
      i < i+1 by XREAL_1:29;
      then p`1 < G*(i+1,j)`1 by A3,A4,A6,A7,A15,A17,Th3;
      then p in { |[r,s]| : G*(i,j)`1 <= r & r <= G*(i+1,j)`1 } by A11,A16;
      hence thesis by A2,A4,A6,A7,A14,Th8;
    end;
  end;
  hence thesis;
end;
