reserve S for Subset of TOP-REAL 2,
  C,C1,C2 for non empty compact Subset of TOP-REAL 2,
  p,q for Point of TOP-REAL 2;
reserve i,j,k for Nat,
  t,r1,r2,s1,s2 for Real;
reserve D1 for non vertical non empty compact Subset of TOP-REAL 2,
  D2 for non horizontal non empty compact Subset of TOP-REAL 2,
  D for non vertical non horizontal non empty compact Subset of TOP-REAL 2;

theorem Th55:
  p`2 <= q`2 implies S-bound LSeg(p,q) = p`2
proof
  assume
A1: p`2 <= q`2;
  then
A2: proj2.:LSeg(p,q) = [.p`2,q`2.] by Th53;
  thus S-bound LSeg(p,q) = lower_bound(proj2.:LSeg(p,q)) by Th44
    .= p`2 by A1,A2,JORDAN5A:19;
end;
