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 Th44:
  S-bound S = lower_bound(proj2.:S)
proof
  thus S-bound S = lower_bound rng(proj2|S) by RELSET_1:22
    .= lower_bound(proj2.:S) by RELAT_1:115;
end;
