
theorem
  for S1,S2 being non empty ManySortedSign for A1 being finite-yielding
  non-empty MSAlgebra over S1 for A2 being finite-yielding non-empty MSAlgebra
  over S2 st the Sorts of A1 tolerates the Sorts of A2 holds A1+*A2 is
  finite-yielding
proof
  let S1,S2 be non empty ManySortedSign;
  let A1 be finite-yielding non-empty MSAlgebra over S1, A2 be finite-yielding
  non-empty MSAlgebra over S2 such that
A1: the Sorts of A1 tolerates the Sorts of A2;
  let x be object;
  set A = A1+*A2;
  set SA = the Sorts of A, SA1 = the Sorts of A1, SA2 = the Sorts of A2;
A2: dom SA1 = the carrier of S1 by PARTFUN1:def 2;
A3: SA1 is finite-yielding by MSAFREE2:def 11;
  assume x in the carrier of S1+*S2;
  then reconsider x as Vertex of S1+*S2;
A4: dom SA2 = the carrier of S2 by PARTFUN1:def 2;
  the carrier of S1+*S2 = (the carrier of S1) \/ the carrier of S2 by Def2;
  then
A5: x in the carrier of S1 or x in the carrier of S2 by XBOOLE_0:def 3;
A6: SA2 is finite-yielding by MSAFREE2:def 11;
  SA = SA1+*SA2 by A1,Def4;
  then
  SA.x = SA1.x & SA1.x is finite or SA.x = SA2.x & SA2.x is finite by A1,A2,A4
,A5,A3,A6,FUNCT_4:13,15;
  hence thesis;
end;
