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

theorem Th3:
  for A being non-empty Circuit of IIG, iv being InputValues of A,
  v being Vertex of IIG, x being Element of (the Sorts of A).v st v in
InputVertices IIG holds (Fix_inp_ext iv).v.(root-tree[x,v]) = root-tree[iv.v,v]
proof
  let A be non-empty Circuit of IIG, iv be InputValues of A, v be Vertex of
  IIG, x be Element of (the Sorts of A).v;
  set e = root-tree[x,v];
  assume
A1: v in InputVertices IIG;
A2: e in FreeGen(v, the Sorts of A) by MSAFREE:def 15;
  Fix_inp iv c= Fix_inp_ext iv by Def2;
  then
A3: (Fix_inp iv).v c= (Fix_inp_ext iv).v;
  FreeEnv A = MSAlgebra (#FreeSort the Sorts of A,FreeOper the Sorts of A
  #) by MSAFREE:def 14;
  then reconsider e as Element of (the Sorts of FreeEnv A).v by A2;
  e in (FreeGen the Sorts of A).v by A2,MSAFREE:def 16;
  then e in dom((Fix_inp iv).v) by FUNCT_2:def 1;
  hence (Fix_inp_ext iv).v.root-tree[x,v] = (Fix_inp iv).v.e by A3,GRFUNC_1:2
    .= (FreeGen(v, the Sorts of A) --> root-tree [iv.v,v]).e by A1,Def1
    .= root-tree[iv.v,v] by A2,FUNCOP_1:7;
end;
