let N be non empty with_non-empty_elements set ; :: thesis: for S being non empty stored-program IC-Ins-separated definite realistic COM-Struct of N
for p being PartState of S holds p = (ProgramPart p) +* (NPP p)
let S be non empty stored-program IC-Ins-separated definite realistic COM-Struct of N; :: thesis: for p being PartState of S holds p = (ProgramPart p) +* (NPP p)
let p be PartState of S; :: thesis: p = (ProgramPart p) +* (NPP p)
A1: dom (NPP p) c= {(IC )} \/ (dom (DataPart p)) by Th171;
A2: not IC in NAT by Def12;
dom (ProgramPart p) c= NAT by RELAT_1:def 18;
then not IC in dom (ProgramPart p) by A2;
then A3: dom (ProgramPart p) misses {(IC )} by ZFMISC_1:56;
dom (ProgramPart p) misses dom (DataPart p) by Th15;
then dom (ProgramPart p) misses {(IC )} \/ (dom (DataPart p)) by A3, XBOOLE_1:70;
then A4: dom (ProgramPart p) misses dom (NPP p) by A1, XBOOLE_1:63;
ProgramPart p c= p by RELAT_1:88;
hence p = p \/ (ProgramPart p) by XBOOLE_1:12
.= (ProgramPart p) \/ (NPP p) by XBOOLE_1:39
.= (ProgramPart p) +* (NPP p) by A4, FUNCT_4:32 ;
:: thesis: verum