
theorem Th28:
  for X being non empty finite set for S1, S2 being Signature of X
  for A1 being Circuit of X,S1 for A2 being Circuit of X,S2 holds A1+*A2 is
  Circuit of S1+*S2
proof
  let X be non empty finite set;
  let S1,S2 be Signature of X;
A1: the carrier of S1+*S2 = (the carrier of S1) \/ the carrier of S2 by
CIRCCOMB:def 2;
  let A1 be Circuit of X,S1;
  let A2 be Circuit of X,S2;
A2: dom the Sorts of A1 = the carrier of S1 & dom the Sorts of A2 = the
  carrier of S2 by PARTFUN1:def 2;
A3: the Sorts of A1 is finite-yielding & the Sorts of A2 is finite-yielding
  by MSAFREE2:def 11;
  A1 tolerates A2 by Th27;
  then
A4: the Sorts of A1 tolerates the Sorts of A2;
  then
A5: the Sorts of A1+*A2 = (the Sorts of A1)+*the Sorts of A2 by CIRCCOMB:def 4;
  A1+*A2 is finite-yielding
  proof
    let i be object;
    assume i in the carrier of S1+*S2;
    then i in the carrier of S1 or i in the carrier of S2 by A1,XBOOLE_0:def 3;
    then
    i in the carrier of S1 & (the Sorts of A1+*A2).i = (the Sorts of A1).
i or i in the carrier of S2 & (the Sorts of A1+*A2).i = (the Sorts of A2).i by
A4,A5,A2,FUNCT_4:13,15;
    hence thesis by A3;
  end;
  hence thesis;
end;
