
theorem Th17:
  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 st s is stable holds (for s1 being State of
A1 st s1 = s|the carrier of S1 holds s1 is stable) & (for s2 being State of A2
  st s2 = s|the carrier of S2 holds s2 is stable)
proof
  let S1,S2,S be non void Circuit-like non empty ManySortedSign such that
A1: S = S1+*S2;
A2: the carrier of S = (the carrier of S1) \/ the carrier of S2 by A1,
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
A3: A1 tolerates A2 & A = A1+*A2;
  let s be State of A such that
A4: s = Following s;
  hereby
    let s1 be State of A1 such that
A5: s1 = s|the carrier of S1;
A6: now
      let x be object;
      assume x in the carrier of S1;
      then reconsider v = x as Vertex of S1;
      reconsider v9 = v as Vertex of S by A2,XBOOLE_0:def 3;
      the carrier of S1 = (InputVertices S1) \/ InnerVertices S1 by XBOOLE_1:45
;
      then v in InputVertices S1 or v9 in InnerVertices S1 by XBOOLE_0:def 3;
      then s1.v = (Following s1).v or s.v9 = (Following s1).v by A1,A3,A4,A5,
CIRCCOMB:31,CIRCUIT2:def 5;
      hence s1.x = (Following s1).x by A5,FUNCT_1:49;
    end;
    dom s1 = the carrier of S1 & dom Following s1 = the carrier of S1 by
CIRCUIT1:3;
    then s1 = Following s1 by A6,FUNCT_1:2;
    hence s1 is stable;
  end;
  let s2 be State of A2 such that
A7: s2 = s|the carrier of S2;
A8: now
    let x be object;
    assume x in the carrier of S2;
    then reconsider v = x as Vertex of S2;
    reconsider v9 = v as Vertex of S by A2,XBOOLE_0:def 3;
    the carrier of S2 = (InputVertices S2) \/ InnerVertices S2 by XBOOLE_1:45;
    then v in InputVertices S2 or v9 in InnerVertices S2 by XBOOLE_0:def 3;
    then s2.v = (Following s2).v or s.v9 = (Following s2).v by A1,A3,A4,A7,
CIRCCOMB:31,CIRCUIT2:def 5;
    hence s2.x = (Following s2).x by A7,FUNCT_1:49;
  end;
  dom s2 = the carrier of S2 & dom Following s2 = the carrier of S2 by
CIRCUIT1:3;
  hence s2 = Following s2 by A8,FUNCT_1:2;
end;
