let IIG be non empty non void Circuit-like monotonic ManySortedSign ; :: thesis: for A being non-empty Circuit of IIG
for iv being InputValues of A
for v being Vertex of IIG
for x being Element of the Sorts of A . v st v in InputVertices IIG holds
(() . v) . (root-tree [x,v]) = root-tree [(iv . v),v]

let A be non-empty Circuit of IIG; :: thesis: for iv being InputValues of A
for v being Vertex of IIG
for x being Element of the Sorts of A . v st v in InputVertices IIG holds
(() . v) . (root-tree [x,v]) = root-tree [(iv . v),v]

let iv be InputValues of A; :: thesis: for v being Vertex of IIG
for x being Element of the Sorts of A . v st v in InputVertices IIG holds
(() . v) . (root-tree [x,v]) = root-tree [(iv . v),v]

let v be Vertex of IIG; :: thesis: for x being Element of the Sorts of A . v st v in InputVertices IIG holds
(() . v) . (root-tree [x,v]) = root-tree [(iv . v),v]

let x be Element of the Sorts of A . v; :: thesis: ( v in InputVertices IIG implies (() . v) . (root-tree [x,v]) = root-tree [(iv . v),v] )
set e = root-tree [x,v];
assume A1: v in InputVertices IIG ; :: thesis: (() . v) . (root-tree [x,v]) = root-tree [(iv . v),v]
A2: root-tree [x,v] 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= () . v ;
FreeEnv A = MSAlgebra(# (FreeSort the Sorts of A),(FreeOper the Sorts of A) #) by MSAFREE:def 14;
then reconsider e = root-tree [x,v] as Element of the Sorts of () . v by A2;
e in (FreeGen the Sorts of A) . v by ;
then e in dom ((Fix_inp iv) . v) by FUNCT_2:def 1;
hence (() . v) . (root-tree [x,v]) = ((Fix_inp iv) . v) . e by
.= ((FreeGen (v, the Sorts of A)) --> (root-tree [(iv . v),v])) . e by
.= root-tree [(iv . v),v] by ;
:: thesis: verum