
theorem Th31:
  for S1,S2,S being non void Circuit-like non empty
  ManySortedSign st S = S1+*S2 for A1 being non-empty Circuit of S1, A2 being
non-empty Circuit of S2 for A being non-empty Circuit of S st A1 tolerates A2 &
  A = A1+*A2 for s being State of A, v being Vertex of S holds (for s1 being
State of A1 st s1 = s|the carrier of S1 holds v in InnerVertices S1 or v in the
carrier of S1 & v in InputVertices S implies (Following s).v = (Following s1).v
  ) & (for s2 being State of A2 st s2 = s|the carrier of S2 holds v in
  InnerVertices S2 or v in the carrier of S2 & v in InputVertices S implies (
  Following s).v = (Following s2).v)
proof
  let S1,S2,S be non void Circuit-like non empty ManySortedSign such that
A1: S = S1+*S2;
  let A1 be non-empty Circuit of S1, A2 be non-empty Circuit of S2;
  let A be non-empty Circuit of S such that
A2: S1 tolerates S2 and
A3: the Sorts of A1 tolerates the Sorts of A2 and
A4: the Charact of A1 tolerates the Charact of A2 and
A5: A = A1+*A2;
  let s be State of A, v be Vertex of S;
  hereby
    let s1 be State of A1 such that
A6: s1 = s|the carrier of S1;
A7: now
      assume v in InnerVertices S1;
      then reconsider v1 = v as Element of InnerVertices S1;
A8:   (Following s1).v1 = (Den (action_at v1, A1)).(action_at v1
      depends_on_in s1) by CIRCUIT2:def 5;
      v1 in InnerVertices S by A1,A2,Th17;
      then
A9:   (Following s).v = (Den (action_at v, A)).(action_at v depends_on_in
      s) by CIRCUIT2:def 5;
A10:  action_at v = action_at v1 by A1,A2,Th17;
      then Den (action_at v1, A1) = Den (action_at v, A) by A1,A3,A4,A5,Th28;
      hence (Following s).v = (Following s1).v by A1,A2,A6,A10,A9,A8,Th30;
    end;
    now
      assume that
A11:  v in the carrier of S1 and
A12:  v in InputVertices S;
      reconsider v1 = v as Vertex of S1 by A11;
      v1 in InputVertices S1 by A1,A2,A12,Th13;
      then
A13:  (Following s1).v1 = s1.v1 by CIRCUIT2:def 5;
A14:  dom s1 = the carrier of S1 by CIRCUIT1:3;
      (Following s).v = s.v by A12,CIRCUIT2:def 5;
      hence (Following s).v = (Following s1).v by A6,A13,A14,FUNCT_1:47;
    end;
    hence v in InnerVertices S1 or v in the carrier of S1 & v in InputVertices
    S implies (Following s).v = (Following s1).v by A7;
  end;
  let s2 be State of A2 such that
A15: s2 = s|the carrier of S2;
A16: now
    assume v in InnerVertices S2;
    then reconsider v2 = v as Element of InnerVertices S2;
A17: (Following s2).v2 = (Den (action_at v2, A2)).(action_at v2
    depends_on_in s2) by CIRCUIT2:def 5;
    v2 in InnerVertices S by A1,Th15;
    then
A18: (Following s).v = (Den (action_at v, A)).(action_at v depends_on_in s
    ) by CIRCUIT2:def 5;
A19: action_at v = action_at v2 by A1,Th15;
    then Den (action_at v2, A2) = Den (action_at v, A) by A1,A3,A5,Th27;
    hence (Following s).v = (Following s2).v by A1,A15,A19,A18,A17,Th29;
  end;
  now
    assume that
A20: v in the carrier of S2 and
A21: v in InputVertices S;
    reconsider v2 = v as Vertex of S2 by A20;
    v2 in InputVertices S2 by A1,A21,Th12;
    then
A22: (Following s2).v2 = s2.v2 by CIRCUIT2:def 5;
A23: dom s2 = the carrier of S2 by CIRCUIT1:3;
    (Following s).v = s.v by A21,CIRCUIT2:def 5;
    hence (Following s).v = (Following s2).v by A15,A22,A23,FUNCT_1:47;
  end;
  hence thesis by A16;
end;
