
theorem Th30:
  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 the Sorts of A1
  +*A2 is constant & the_value_of the Sorts of A1+*A2 = X
proof
  let X be non empty finite set;
  let S1,S2 be Signature of X;
  let A1 be Circuit of X,S1;
  let A2 be Circuit of X,S2;
  reconsider A = A1+*A2 as Circuit of S1+*S2 by Th28;
A1: dom the Sorts of A1 = the carrier of S1 by PARTFUN1:def 2;
  the Sorts of A1 is constant & the_value_of the Sorts of A1 = X by Def10;
  then the Sorts of A1 = (dom the Sorts of A1)-->X by FUNCOP_1:80;
  then
A2: the Sorts of A1 = [:dom the Sorts of A1,{X}:] by FUNCOP_1:def 2;
  the Sorts of A2 is constant & the_value_of the Sorts of A2 = X by Def10;
  then the Sorts of A2 = (dom the Sorts of A2)-->X by FUNCOP_1:80;
  then
A3: the Sorts of A2 = [:dom the Sorts of A2,{X}:] by FUNCOP_1:def 2;
  A1 tolerates A2 by Th27;
  then
A4: the Sorts of A1 tolerates the Sorts of A2;
  then the Sorts of A = (the Sorts of A1)+*(the Sorts of A2) by CIRCCOMB:def 4
    .= (the Sorts of A1)\/(the Sorts of A2) by A4,FUNCT_4:30
    .= [:(dom the Sorts of A1) \/ dom the Sorts of A2, {X}:] by A2,A3,
ZFMISC_1:97
    .= ((the carrier of S1) \/ dom the Sorts of A2) --> X by A1,FUNCOP_1:def 2;
  hence thesis by FUNCOP_1:79;
end;
