
theorem Th48:
  for S1,S2 being non empty ManySortedSign, A1 being MSAlgebra
  over S1 for A2 being MSAlgebra over S2 st A1 is gate`2=den & A2 is gate`2=den
  holds the Charact of A1 tolerates the Charact of A2
proof
  let S1,S2 be non empty ManySortedSign;
  let A1 be MSAlgebra over S1, A2 be MSAlgebra over S2 such that
A1: A1 is gate`2=den and
A2: A2 is gate`2=den;
  let x be object;
  set C1 = the Charact of A1, C2 = the Charact of A2;
  assume
A3: x in dom C1 /\ dom C2;
  then x in dom C2 by XBOOLE_0:def 4;
  then x in the carrier' of S2 by PARTFUN1:def 2;
  then
A4: x = [x`1, C2.x] by A2;
  x in dom C1 by A3,XBOOLE_0:def 4;
  then x in the carrier' of S1 by PARTFUN1:def 2;
  then x = [x`1, C1.x] by A1;
  hence thesis by A4,XTUPLE_0:1;
end;
