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 Th48:
  S = C1 \/ C2 implies S-bound S = min(S-bound C1, S-bound C2)
proof
  assume
A1: S = C1 \/ C2;
A2: S-bound C1 = lower_bound(proj2.:C1) by Th44;
A3: S-bound C2 = lower_bound(proj2.:C2) by Th44;
  thus S-bound S = lower_bound(proj2.:S) by Th44
    .= lower_bound(proj2.:C1 \/ proj2.:C2) by A1,RELAT_1:120
    .= min(S-bound C1, S-bound C2) by A2,A3,SEQ_4:142;
end;
