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 holds dom (ProgramPart s) = NAT
let S be non empty stored-program COM-Struct of N; :: thesis: for s being State of S holds dom (ProgramPart s) = NAT
let s be State of S; :: thesis: dom (ProgramPart s) = NAT
A1: NAT c= the carrier of S by Def3;
dom s = the carrier of S by PARTFUN1:def 4;
hence dom (ProgramPart s) = NAT by A1, RELAT_1:91; :: thesis: verum