
theorem
  for S1,S2 being Circuit-like non empty ManySortedSign st
  InnerVertices S1 misses InnerVertices S2 holds S1+*S2 is Circuit-like
proof
  let S1,S2 be Circuit-like non empty ManySortedSign;
  assume
A1: InnerVertices S1 misses InnerVertices S2;
  set r1 = the ResultSort of S1, r2 = the ResultSort of S2;
  let S be non void non empty ManySortedSign;
  set r = the ResultSort of S;
A2: dom r1 = the carrier' of S1 by FUNCT_2:def 1;
  assume
A3: S = S1+*S2;
  then
A4: r = r1+*r2 by Def2;
A5: dom r2 = the carrier' of S2 by FUNCT_2:def 1;
  let o1, o2 be Gate of S;
A6: the carrier' of S = (the carrier' of S1) \/ the carrier' of S2 by A3,Def2;
  then not o1 in the carrier' of S2 & o1 in the carrier' of S1 or o1 in the
  carrier' of S2 by XBOOLE_0:def 3;
  then
A7: r.o1 = r1.o1 & r1.o1 in rng r1 & o1 in the carrier' of S1 or r.o1 = r2.
  o1 & r2.o1 in rng r2 & o1 in the carrier' of S2 by A4,A2,A5,A6,FUNCT_1:def 3
,FUNCT_4:def 1;
  not o2 in the carrier' of S2 & o2 in the carrier' of S1 or o2 in the
  carrier' of S2 by A6,XBOOLE_0:def 3;
  then
A8: r.o2 = r1.o2 & r1.o2 in rng r1 & o2 in the carrier' of S1 or r.o2 = r2.
  o2 & r2.o2 in rng r2 & o2 in the carrier' of S2 by A4,A2,A5,A6,FUNCT_1:def 3
,FUNCT_4:def 1;
  assume
A9: the_result_sort_of o1 = the_result_sort_of o2;
  per cases by A1,A9,A7,A8,XBOOLE_0:3;
  suppose
A10: r.o1 = r1.o1 & o1 in the carrier' of S1 & r.o2 = r1.o2 & o2 in
    the carrier' of S1;
    then reconsider
    S = S1 as non void Circuit-like non empty ManySortedSign;
    reconsider p1 = o1, p2 = o2 as Gate of S by A10;
A11: the_result_sort_of p2 = r1.p2;
    the_result_sort_of p1 = r1.p1;
    hence thesis by A9,A10,A11,MSAFREE2:def 6;
  end;
  suppose
A12: r.o1 = r2.o1 & o1 in the carrier' of S2 & r.o2 = r2.o2 & o2 in
    the carrier' of S2;
    then reconsider
    S = S2 as non void Circuit-like non empty ManySortedSign;
    reconsider p1 = o1, p2 = o2 as Gate of S by A12;
A13: the_result_sort_of p2 = r2.p2;
    the_result_sort_of p1 = r2.p1;
    hence thesis by A9,A12,A13,MSAFREE2:def 6;
  end;
end;
