
theorem Th63:
  for S1,S2 being non empty ManySortedSign for A1 be non-empty
  MSAlgebra over S1, A2 be non-empty MSAlgebra over S2 st A1 is gate`2=den & A2
  is gate`2=den & the Sorts of A1 tolerates the Sorts of A2 holds A1+*A2 is
  gate`2=den
proof
  let S1,S2 be non empty ManySortedSign;
  let A1 be non-empty MSAlgebra over S1;
  let A2 be non-empty MSAlgebra over S2;
  set A = A1+*A2;
  set S = S1+*S2;
  assume that
A1: A1 is gate`2=den and
A2: A2 is gate`2=den and
A3: the Sorts of A1 tolerates the Sorts of A2;
A4: the Charact of A = (the Charact of A1)+*the Charact of A2 by A3,Def4;
  let g be set;
A5: dom the Charact of A1 = the carrier' of S1 by PARTFUN1:def 2;
A6: dom the Charact of A2 = the carrier' of S2 by PARTFUN1:def 2;
A7: the carrier' of S = (the carrier' of S1) \/ the carrier' of S2 by Def2;
  assume g in the carrier' of S;
  then
A8: g in the carrier' of S1 or g in the carrier' of S2 by A7,XBOOLE_0:def 3;
  the Charact of A1 tolerates the Charact of A2 by A1,A2,Th48;
  then
  (the Charact of A).g = (the Charact of A1).g & [g`1, (the Charact of A1
  ).g] = g or (the Charact of A).g = (the Charact of A2).g & [g`1, (the Charact
  of A2).g] = g by A1,A2,A4,A5,A6,A8,FUNCT_4:13,15;
  hence thesis;
end;
