let N be with_non-empty_elements set ; :: thesis: for S being non empty stored-program COM-Struct of N
for s being State of S
for l being Element of NAT holds l in dom s
let S be non empty stored-program COM-Struct of N; :: thesis: for s being State of S
for l being Element of NAT holds l in dom s
let s be State of S; :: thesis: for l being Element of NAT holds l in dom s
let l be Element of NAT ; :: thesis: l in dom s
X: l in NAT ;
NAT c= the carrier of S by Def3;
then l in the carrier of S by X;
hence l in dom s by PARTFUN1:def 4; :: thesis: verum