
theorem Th17:
  for S1,S being Circuit-like non void non empty ManySortedSign,
S2 being non empty ManySortedSign st S1 tolerates S2 & S = S1+*S2 for v1 being
Vertex of S1 st v1 in InnerVertices S1 for v being Vertex of S st v1 = v holds
  v in InnerVertices S & action_at v = action_at v1
proof
  let S1,S be Circuit-like non void non empty ManySortedSign, S2 be non
  empty ManySortedSign such that
A1: S1 tolerates S2 and
A2: S = S1+*S2;
  let v1 be Vertex of S1 such that
A3: v1 in InnerVertices S1;
  let v be Vertex of S such that
A4: v1 = v;
  InnerVertices S = (InnerVertices S1) \/ (InnerVertices S2) by A1,A2,Th11;
  hence
A5: v in InnerVertices S by A3,A4,XBOOLE_0:def 3;
  the carrier' of S = (the carrier' of S1) \/ the carrier' of S2 by A2,Def2;
  then reconsider o = action_at v1 as OperSymbol of S by XBOOLE_0:def 3;
  the_result_sort_of action_at v1 = v1 by A3,MSAFREE2:def 7;
  then v = the_result_sort_of o by A1,A2,A4,Th16;
  hence thesis by A5,MSAFREE2:def 7;
end;
