
theorem Th62:
  for S1,S2 being non empty ManySortedSign for A1 be Boolean
  MSAlgebra over S1 for A2 be Boolean MSAlgebra over S2 holds A1+*A2 is Boolean
proof
  let S1,S2 be non empty ManySortedSign;
  let A1 be Boolean MSAlgebra over S1;
  let A2 be Boolean MSAlgebra over S2;
  set A = A1+*A2;
  set S = S1+*S2;
  let x be Vertex of S;
  the carrier of S = (the carrier of S1) \/ the carrier of S2 by Def2;
  then
A3: x in the carrier of S1 or x in the carrier of S2 by XBOOLE_0:def 3;
A4: the Sorts of A2 = (the carrier of S2) --> BOOLEAN by Th57;
A5: the Sorts of A1 = (the carrier of S1) --> BOOLEAN by Th57;
  then
A6: the Sorts of A1 tolerates the Sorts of A2 by A4,FUNCOP_1:87;
  then the Sorts of A = (the Sorts of A1)+*the Sorts of A2 by Def4;
  then
  (the Sorts of A).x = (the Sorts of A1).x & (the Sorts of A1).x = BOOLEAN
or (the Sorts of A).x = (the Sorts of A2).x & (the Sorts of A2).x = BOOLEAN by
A5,A4,A6,A3,FUNCOP_1:7,FUNCT_4:13,15;
  hence thesis;
end;
