
theorem Th15:
  for S1,S2,S being non void Circuit-like non empty
ManySortedSign st InputVertices S1 misses InnerVertices S2 & 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
  for s1 being State of A1 st s1 = s|the carrier of S1 & s1 is stable for s2
  being State of A2 st s2 = s|the carrier of S2 holds (Following s)|the carrier
  of S2 = Following s2
proof
  let S1,S2,S be non void Circuit-like non empty ManySortedSign such that
A1: InputVertices S1 misses InnerVertices S2 and
A2: S = S1+*S2;
A3: the carrier of S = (the carrier of S1)\/the carrier of S2 by A2,
CIRCCOMB:def 2;
  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
A4: A1 tolerates A2 and
A5: A = A1+*A2;
  let s be State of A;
  let s1 be State of A1 such that
A6: s1 = s|the carrier of S1 and
A7: s1 is stable;
  let s2 be State of A2 such that
A8: s2 = s|the carrier of S2;
A9: now
    let a be object;
    assume a in the carrier of S2;
    then reconsider v = a as Vertex of S2;
    reconsider vv = v as Vertex of S by A3,XBOOLE_0:def 3;
A10: v in InputVertices S2 & not v in InnerVertices S1 implies v in (
    InputVertices S2)\InnerVertices S1 by XBOOLE_0:def 5;
A11: S1 tolerates S2 by A4,CIRCCOMB:def 3;
A12: now
      assume that
A13:  v in InputVertices S2 and
A14:  v in InnerVertices S1;
      reconsider v1 = v as Vertex of S1 by A14;
      thus (Following s).vv = (Following s1).vv by A2,A4,A5,A6,A14,CIRCCOMB:31
        .= s1.v by A7
        .= s.v1 by A6,FUNCT_1:49
        .= s2.v by A8,FUNCT_1:49
        .= (Following s2).vv by A13,CIRCUIT2:def 5;
    end;
    the carrier of S2 = InnerVertices S2 \/ InputVertices S2 & (
InputVertices S1 )\InnerVertices S2 = InputVertices S1 by A1,XBOOLE_1:45,83;
    then v in InnerVertices S2 or v in InputVertices S2 & (v in InnerVertices
    S1 or not v in InnerVertices S1) & InputVertices S = (InputVertices S1)\/((
    InputVertices S2)\InnerVertices S1) by A2,A11,Th5,XBOOLE_0:def 3;
    then
A15: vv in InnerVertices S2 or v in InputVertices S or v in InputVertices
    S2 & v in InnerVertices S1 by A10,XBOOLE_0:def 3;
    thus ((Following s)|the carrier of S2).a = (Following s).v by FUNCT_1:49
      .= (Following s2).a by A2,A4,A5,A8,A12,A15,CIRCCOMB:31;
  end;
  dom Following s = the carrier of S by CIRCUIT1:3;
  then dom Following s2 = the carrier of S2 & dom ((Following s)|the carrier
  of S2) = the carrier of S2 by A3,CIRCUIT1:3,RELAT_1:62,XBOOLE_1:7;
  hence thesis by A9,FUNCT_1:2;
end;
