reserve IIG for monotonic Circuit-like non void non empty ManySortedSign;

theorem Th12:
  for SCS being non-empty Circuit of IIG, s being State of SCS, iv
  being InputValues of SCS st iv c= s holds iv c= Following s
proof
  let SCS be non-empty Circuit of IIG, s be State of SCS, iv be InputValues of
  SCS such that
A1: iv c= s;
  now
    dom s = the carrier of IIG by CIRCUIT1:3
      .= dom Following s by CIRCUIT1:3;
    hence dom iv c= dom Following s by A1,RELAT_1:11;
    let x be object such that
A2: x in dom iv;
A3: dom iv = InputVertices IIG by PARTFUN1:def 2;
    then reconsider v = x as Vertex of IIG by A2;
    iv.v = s.v by A1,A2,GRFUNC_1:2;
    hence iv.x = (Following s).x by A2,A3,Def5;
  end;
  hence thesis by GRFUNC_1:2;
end;
