
theorem Th10:
  for S1,S2 being non void Circuit-like non empty ManySortedSign
st InputVertices S1 misses InnerVertices S2 for S being non void Circuit-like
  non empty ManySortedSign st S = S1+*S2 for A1 being non-empty Circuit of S1
  for A2 being non-empty Circuit of S2 st A1 tolerates A2 for A being non-empty
Circuit of S st A = A1 +* A2 for s being State of A for s1 being State of A1 st
  s1 = s|the carrier of S1 & s1 is stabilizing for s2 being State of A2 st s2 =
Following(s, stabilization-time s1)|the carrier of S2 & s2 is stabilizing holds
  stabilization-time(s) = (stabilization-time s1)+(stabilization-time s2)
proof
  let S1,S2 be non void Circuit-like non empty ManySortedSign such that
A1: InputVertices S1 misses InnerVertices S2;
  let S be non void Circuit-like non empty ManySortedSign such that
A2: S=S1 +* S2;
  let A1 be non-empty Circuit of S1;
  let A2 be non-empty Circuit of S2;
  assume
A3: A1 tolerates A2;
  let A be non-empty Circuit of S such that
A4: A = A1 +* A2;
  let s be State of A;
  let s1 be State of A1 such that
A5: s1=s|the carrier of S1 and
A6: s1 is stabilizing;
  set st1=stabilization-time(s1);
  let s2 be State of A2 such that
A7: s2=Following(s,st1)|the carrier of S2 and
A8: s2 is stabilizing;
  set st2=stabilization-time(s2);
A9: Following(s1,st1) is stable by A6,Def5;
A10: now
    let n be Element of NAT such that
A11: n < st1+st2;
    per cases;
    suppose
      st1 <= n;
      then consider m being Nat such that
A12:  n = st1+m by NAT_1:10;
      reconsider m as Element of NAT by ORDINAL1:def 12;
      m < st2 by A11,A12,XREAL_1:6;
      then
A13:  Following(s2,m) is not stable by A8,Def5;
      Following(s1,st1) = Following(s,st1)|the carrier of S1 by A1,A2,A3,A4,A5,
CIRCCMB2:13;
      then Following(s2,m) = Following(Following(s,st1),m)|the carrier of S2
      by A1,A2,A3,A4,A7,A9,CIRCCMB2:18
        .= Following(s,n)|the carrier of S2 by A12,FACIRC_1:13;
      hence Following(s,n) is not stable by A2,A3,A4,A13,CIRCCMB2:17;
    end;
    suppose
      n < st1;
      then
A14:  Following(s1,n) is not stable by A6,Def5;
      Following(s,n)|the carrier of S1 = Following(s1,n) by A1,A2,A3,A4,A5,
CIRCCMB2:13;
      hence Following(s,n) is not stable by A2,A3,A4,A14,CIRCCMB2:17;
    end;
  end;
  Following(s2,st2) is stable by A8,Def5;
  then
A15: Following(s,st1+st2) is stable by A1,A2,A3,A4,A5,A7,A9,CIRCCMB2:19;
  s is stabilizing by A1,A2,A3,A4,A5,A6,A7,A8,Th9;
  hence thesis by A15,A10,Def5;
end;
