reserve C for Simple_closed_curve,
  i, j, n for Nat,
  p for Point of TOP-REAL 2;

theorem
  for C1 being non empty compact Subset of TOP-REAL 2, C2, S being non
  empty Subset of TOP-REAL 2 holds S = C1 \/ C2 & proj1.:C2 is non empty
  bounded_below implies W-bound S = min(W-bound C1, W-bound C2)
proof
  let C1 be non empty compact Subset of TOP-REAL 2, C2, S be non empty Subset
  of TOP-REAL 2;
  assume that
A1: S = C1 \/ C2 and
A2: proj1.:C2 is non empty bounded_below;
  set P1 = proj1.:C1, P2 = proj1.:C2, PS = proj1.:S;
A3: W-bound C1 = lower_bound P1 by SPRECT_1:43;
A4: W-bound C2 = lower_bound P2 by SPRECT_1:43;
  thus W-bound S = lower_bound PS by SPRECT_1:43
    .= lower_bound(P1 \/ P2) by A1,RELAT_1:120
    .= min(W-bound C1, W-bound C2) by A2,A3,A4,SEQ_4:142;
end;
