
theorem
  for S1,S2 being non void Circuit-like non empty ManySortedSign st
  InputVertices S1 misses InnerVertices S2 & InputVertices S2 misses
  InnerVertices S1 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=s|the carrier of S2 & s2 is
  stabilizing holds stabilization-time(s) = max (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 & InputVertices S2 misses
  InnerVertices S1;
  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=s|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 < max(st1,st2);
    per cases;
    suppose
      st1<=st2;
      then n < st2 by A11,XXREAL_0:def 10;
      then not Following(s2,n) is stable by A8,Def5;
      hence not Following(s,n) is stable by A1,A2,A3,A4,A5,A7,CIRCCMB2:23;
    end;
    suppose
      st2<=st1;
      then n < st1 by A11,XXREAL_0:def 10;
      then not Following(s1,n) is stable by A6,Def5;
      hence not Following(s,n) is stable by A1,A2,A3,A4,A5,A7,CIRCCMB2:23;
    end;
  end;
  Following(s2,st2) is stable by A8,Def5;
  then
A12: Following(s,max(st1,st2)) is stable by A1,A2,A3,A4,A5,A7,A9,CIRCCMB2:22;
  s is stabilizing by A1,A2,A3,A4,A5,A6,A7,A8,Th7;
  hence thesis by A12,A10,Def5;
end;
