let N be with_zero set ; :: thesis: for S being non empty with_non-empty_values IC-Ins-separated Mem-Struct over N
for p being PartState of S st IC in dom p holds
p = (DataPart p) +* (Start-At ((IC p),S))
let S be non empty with_non-empty_values IC-Ins-separated Mem-Struct over N; :: thesis: for p being PartState of S st IC in dom p holds
p = (DataPart p) +* (Start-At ((IC p),S))
let p be PartState of S; :: thesis: ( IC in dom p implies p = (DataPart p) +* (Start-At ((IC p),S)) )
A2: dom (DataPart p) misses dom (Start-At ((IC p),S)) by Th3, ZFMISC_1:50;
A3: dom (Start-At ((IC p),S)) misses dom (DataPart p) by Th3, ZFMISC_1:50;
assume IC in dom p ; :: thesis: p = (DataPart p) +* (Start-At ((IC p),S))
then p = (Start-At ((IC p),S)) +* (DataPart p) by Th19;
then A4: p = (Start-At ((IC p),S)) \/ (DataPart p) by A3, FUNCT_4:31;
thus p = (DataPart p) +* (Start-At ((IC p),S)) by A2, A4, FUNCT_4:31; :: thesis: verum