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 Th46:
  E-bound S = upper_bound(proj1.:S)
proof
  thus E-bound S = upper_bound rng(proj1|S) by RELSET_1:22
    .= upper_bound(proj1.:S) by RELAT_1:115;
end;
