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 e being Element of the Sorts of (FreeEnv A) . v
for x being set st v in (InnerVertices IIG) \ (SortsWithConstants IIG) & e = root-tree [x,v] holds
((Fix_inp_ext iv) . v) . e = e
let A be non-empty Circuit of IIG; :: thesis: for iv being InputValues of A
for v being Vertex of IIG
for e being Element of the Sorts of (FreeEnv A) . v
for x being set st v in (InnerVertices IIG) \ (SortsWithConstants IIG) & e = root-tree [x,v] holds
((Fix_inp_ext iv) . v) . e = e
let iv be InputValues of A; :: thesis: for v being Vertex of IIG
for e being Element of the Sorts of (FreeEnv A) . v
for x being set st v in (InnerVertices IIG) \ (SortsWithConstants IIG) & e = root-tree [x,v] holds
((Fix_inp_ext iv) . v) . e = e
let v be Vertex of IIG; :: thesis: for e being Element of the Sorts of (FreeEnv A) . v
for x being set st v in (InnerVertices IIG) \ (SortsWithConstants IIG) & e = root-tree [x,v] holds
((Fix_inp_ext iv) . v) . e = e
let e be Element of the Sorts of (FreeEnv A) . v; :: thesis: for x being set st v in (InnerVertices IIG) \ (SortsWithConstants IIG) & e = root-tree [x,v] holds
((Fix_inp_ext iv) . v) . e = e
let x be set ; :: thesis: ( v in (InnerVertices IIG) \ (SortsWithConstants IIG) & e = root-tree [x,v] implies ((Fix_inp_ext iv) . v) . e = e )
assume that
A1: v in (InnerVertices IIG) \ (SortsWithConstants IIG) and
A2: e = root-tree [x,v] ; :: thesis: ((Fix_inp_ext iv) . v) . e = e
A3: e . {} = [x,v] by A2, TREES_4:3;

A4: now :: thesis: for o being OperSymbol of IIG holds
( not [o, the carrier of IIG] = e . {} or not the_result_sort_of o = v )

given o being OperSymbol of IIG such that A5: [o, the carrier of IIG] = e . {} and
the_result_sort_of o = v ; :: thesis: contradiction
v = the carrier of IIG by A3, A5, XTUPLE_0:1;
hence contradiction by Lm1; :: thesis: verum

end;

set X = the Sorts of A;
FreeEnv A = MSAlgebra(# (FreeSort the Sorts of A),(FreeOper the Sorts of A) #) by MSAFREE:def 14;
then e in (FreeSort the Sorts of A) . v ;
then A6: e in FreeSort ( the Sorts of A,v) by MSAFREE:def 11;
Fix_inp iv c= Fix_inp_ext iv by Def2;
then A7: (Fix_inp iv) . v c= (Fix_inp_ext iv) . v ;
FreeSort ( the Sorts of A,v) = { a where a is Element of TS (DTConMSA the Sorts of A) : ( ex x being set st
( x in the Sorts of A . v & a = root-tree [x,v] ) or ex o being OperSymbol of IIG st
( [o, the carrier of IIG] = a . {} & the_result_sort_of o = v ) ) } by MSAFREE:def 10;
then ex a being Element of TS (DTConMSA the Sorts of A) st
( a = e & ( ex x being set st
( x in the Sorts of A . v & a = root-tree [x,v] ) or ex o being OperSymbol of IIG st
( [o, the carrier of IIG] = a . {} & the_result_sort_of o = v ) ) ) by A6;
then A8: e in FreeGen (v, the Sorts of A) by A4, MSAFREE:def 15;
then e in (FreeGen the Sorts of A) . v by MSAFREE:def 16;
then e in dom ((Fix_inp iv) . v) by FUNCT_2:def 1;
hence ((Fix_inp_ext iv) . v) . e = ((Fix_inp iv) . v) . e by A7, GRFUNC_1:2
.= (id (FreeGen (v, the Sorts of A))) . e by A1, Def1
.= e by A8, FUNCT_1:17 ;
:: thesis: verum