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 Th5:
  G is Y_equal-in-column & 1 <= j & j < width G & 1 <= i & i <= len G
  implies h_strip(G,j) = { |[r,s]| : G*(i,j)`2 <= s & s <= G*(i,j+1)`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;
A6: 1 <= j+1 by A2,NAT_1:13;
A7: j+1 <= width G by A3,NAT_1:13;
A8: G*(i,j)`2 = G*(1,j)`2 by A1,A2,A3,A4,A5,Th1;
  G*(i,j+1)`2 = G*(1,j+1)`2 by A1,A4,A5,A6,A7,Th1;
  hence thesis by A2,A3,A8,Def2;
end;
