
theorem Th15:
  for S1 being non empty ManySortedSign, S2,S being Circuit-like
non void non empty ManySortedSign st S = S1+*S2 for v2 being Vertex of S2 st
  v2 in InnerVertices S2 for v being Vertex of S st v2 = v holds v in
  InnerVertices S & action_at v = action_at v2
proof
  let S1 be non empty ManySortedSign, S2,S be Circuit-like non void non empty
  ManySortedSign such that
A1: S = S1+*S2;
  let v2 be Vertex of S2 such that
A2: v2 in InnerVertices S2;
  the carrier' of S = (the carrier' of S1) \/ the carrier' of S2 by A1,Def2;
  then reconsider o = action_at v2 as OperSymbol of S by XBOOLE_0:def 3;
  let v be Vertex of S such that
A3: v2 = v;
  the ResultSort of S = (the ResultSort of S1)+*the ResultSort of S2 by A1,Def2
;
  then
A4: InnerVertices S2 c= InnerVertices S by FUNCT_4:18;
  hence v in InnerVertices S by A2,A3;
  the_result_sort_of action_at v2 = v2 by A2,MSAFREE2:def 7;
  then v = the_result_sort_of o by A1,A3,Th14;
  hence thesis by A2,A3,A4,MSAFREE2:def 7;
end;
