reserve I, G, H for set, i, x for object,
  A, B, M for ManySortedSet of I,
  sf, sg, sh for Subset-Family of I,
  v, w for Subset of I,
  F for ManySortedFunction of I;
reserve X, Y, Z for ManySortedSet of I;
reserve SF, SG, SH for MSSubsetFamily of M,
  SFe for non-empty MSSubsetFamily of M,
  V, W for ManySortedSubset of M;

theorem Th40:
  union SF c= M
proof
A1: for x being set
for F be Subset-Family of x holds union F is Subset of x;
  let i such that
A2: i in I;
  SF.i is Subset-Family of M.i by A2,Th32;
  then union (SF.i) is Subset of M.i by A1;
  then union (SF.i) c= M.i;
  hence thesis by A2,MBOOLEAN:def 2;
end;
