reserve S for non void non empty ManySortedSign,
  U0 for MSAlgebra over S;

theorem Th4:
  for S be non void non empty ManySortedSign, X be ManySortedSet of
  the carrier of S holds Union coprod X misses [:the carrier' of S,{the carrier
  of S}:]
proof
  let S be non void non empty ManySortedSign, X be ManySortedSet of the
  carrier of S;
  assume Union (coprod X) /\ [:the carrier' of S,{the carrier of S}:] <> {};
  then consider x be object such that
A1: x in Union (coprod X) /\ [:the carrier' of S,{the carrier of S}:] by
XBOOLE_0:def 1;
  x in Union (coprod X) by A1,XBOOLE_0:def 4;
  then x in union rng (coprod X) by CARD_3:def 4;
  then consider A be set such that
A2: x in A and
A3: A in rng (coprod X) by TARSKI:def 4;
  consider s be object such that
A4: s in dom (coprod X) and
A5: (coprod X).s = A by A3,FUNCT_1:def 3;
  reconsider s as SortSymbol of S by A4;
  A = coprod(s,X) by A5,Def3;
  then
A6: ex a be set st a in X.s & x = [a,s] by A2,Def2;
  x in [:the carrier' of S,{the carrier of S}:] by A1,XBOOLE_0:def 4;
  then s in {the carrier of S} by A6,ZFMISC_1:87;
  then s in the carrier of S & s = the carrier of S by TARSKI:def 1;
  hence contradiction;
end;
