
theorem Th19:
  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 n1,n2 being Nat for
s being State of A for s1 being State of A1, s2 being State of A2 st s1 = s|the
carrier of S1 & Following(s1, n1) is stable & s2 = Following(s, n1)|the carrier
  of S2 & Following(s2, n2) is stable holds Following(s, n1+n2) is stable
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;
  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 and
A4: A = A1+*A2;
  let n1,n2 be Nat;
  let s be State of A, s9 be State of A1, s99 be State of A2 such that
A5: s9 = s|the carrier of S1 & Following(s9, n1) is stable and
A6: s99 = Following(s, n1)|the carrier of S2 & Following(s99, n2) is stable;
A7: the Sorts of A1 tolerates the Sorts of A2 by A3,CIRCCOMB:def 3;
  then reconsider
  s1 = Following(s, n1)|the carrier of S1, s0 = s|the carrier of S1
  as State of A1 by A4,CIRCCOMB:26;
A8: Following(Following(s, n1), n2) = Following(s, n1+n2) by FACIRC_1:13;
  then
A9: Following(s, n1+n2)|the carrier of S1 = Following(s1, n2) by A1,A2,A3,A4
,Th13;
  reconsider s2 = Following(s, n1)|the carrier of S2 as State of A2 by A4,A7,
CIRCCOMB:26;
A10: dom Following(s, n1+n2) = the carrier of S by CIRCUIT1:3;
A11: the carrier of S = (the carrier of S1)\/the carrier of S2 by A2,
CIRCCOMB:def 2;
A12: s1 = Following(s0, n1) by A1,A2,A3,A4,Th13;
  then
A13: Following(s, n1+n2)|the carrier of S2 = Following(s2, n2) by A1,A2,A3,A4
,A5,A8,Th18;
  then Following Following(s, n1+n2) = (Following Following(s2, n2))+*
  Following Following(s1, n2) by A1,A2,A3,A4,A9,CIRCCOMB:33
    .= Following(s2, n2)+*Following Following(s1, n2) by A6
    .= Following(s2, n2)+*Following(s1, n2+1) by FACIRC_1:12
    .= Following(s2, n2)+*s1 by A5,A12,Th3
    .= Following(s2, n2)+*Following(s1, n2) by A5,A12,Th3
    .= Following(s, n1+n2) by A11,A10,A9,A13,FUNCT_4:70;
  hence thesis;
end;
