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 Th50:
  S = C1 \/ C2 implies E-bound S = max(E-bound C1, E-bound C2)
proof
  assume
A1: S = C1 \/ C2;
A2: E-bound C1 = upper_bound(proj1.:C1) by Th46;
A3: E-bound C2 = upper_bound(proj1.:C2) by Th46;
  thus E-bound S = upper_bound(proj1.:S) by Th46
    .= upper_bound(proj1.:C1 \/ proj1.:C2) by A1,RELAT_1:120
    .= max(E-bound C1, E-bound C2) by A2,A3,SEQ_4:143;
end;
