
theorem
  for S1,S2 being non empty ManySortedSign for A1 being non-empty
  MSAlgebra over S1, A2 being non-empty MSAlgebra over S2 st the Sorts of A1
  tolerates the Sorts of A2 for s being Element of product the Sorts of A1+*A2
  holds s|the carrier of S1 in product the Sorts of A1 & s|the carrier of S2 in
  product the Sorts of A2
proof
  let S1,S2 be non empty ManySortedSign;
  let A1 be non-empty MSAlgebra over S1;
  let A2 be non-empty MSAlgebra over S2;
A1: dom the Sorts of A1 = the carrier of S1 by PARTFUN1:def 2;
A2: dom the Sorts of A2 = the carrier of S2 by PARTFUN1:def 2;
  assume
A3: the Sorts of A1 tolerates the Sorts of A2;
  then the Sorts of A1+*A2 = (the Sorts of A1)+*the Sorts of A2 by Def4;
  hence thesis by A3,A1,A2,CARD_3:98,99;
end;
