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 Th20:
  G is X_equal-in-line X_increasing-in-column &
  1 <= j & j <= width G & 1 <= i & i+1 <= len G
  implies LSeg(G*(i,j),G*(i+1,j)) c= v_strip(G,i)
proof
  assume that
A1: G is X_equal-in-line and
A2: G is X_increasing-in-column and
A3: 1 <= j and
A4: j <= width G and
A5: 1 <= i and
A6: i+1 <= len G;
  let x be object;
  assume
A7: x in LSeg(G*(i,j),G*(i+1,j));
  then reconsider p = x as Point of TOP-REAL 2;
A8: p = |[p`1, p`2]| by EUCLID:53;
A9: i < len G by A6,NAT_1:13;
  i < i+1 by XREAL_1:29;
  then
A10: G*(i,j)`1 < G*(i+1,j)`1 by A2,A3,A4,A5,A6,Th3;
  then
A11: G*(i,j)`1 <= p`1 by A7,TOPREAL1:3;
  p`1 <= G*(i+1,j)`1 by A7,A10,TOPREAL1:3;
  then p in { |[r,s]| : G*(i,j)`1 <= r & r <= G*(i+1,j)`1 } by A8,A11;
  hence thesis by A1,A3,A4,A5,A9,Th8;
end;
