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

theorem Th2:
  for IIG for SCS being non-empty Circuit of IIG, InpFs being
InputFuncs of SCS, n being Nat st IIG is with_input_V holds (commute
  InpFs).n is InputValues of SCS
proof
  let IIG;
  let SCS be non-empty Circuit of IIG, InpFs be InputFuncs of SCS, n be Nat;
  reconsider A = InputVertices IIG as Subset of IIG;
  reconsider SS = the Sorts of SCS as non-empty ManySortedSet of the carrier
  of IIG;
  assume IIG is with_input_V;
  then reconsider A as non empty Subset of IIG;
  reconsider SI = SS | A as ManySortedSet of A;
  reconsider SI as non-empty ManySortedSet of A;
  reconsider n as Element of NAT by ORDINAL1:def 12;
  consider ivm being ManySortedSet of A such that
A1: ivm = (commute InpFs).n and
A2: ivm in SI by PRE_CIRC:7;
  now
    let v be Vertex of IIG;
    assume
A3: v in InputVertices IIG;
    then SI.v = (the Sorts of SCS).v by FUNCT_1:49;
    hence ivm.v in (the Sorts of SCS).v by A2,A3,PBOOLE:def 1;
  end;
  hence thesis by A1,MSAFREE2:def 5;
end;
