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
for k being Nat st IC in dom p holds
DecIC (p,k) = (DataPart p) +* (Start-At (((IC p) -' k),S))
let S be non empty with_non-empty_values IC-Ins-separated Mem-Struct over N; :: thesis: for p being PartState of S
for k being Nat st IC in dom p holds
DecIC (p,k) = (DataPart p) +* (Start-At (((IC p) -' k),S))
let p be PartState of S; :: thesis: for k being Nat st IC in dom p holds
DecIC (p,k) = (DataPart p) +* (Start-At (((IC p) -' k),S))
let k be Nat; :: thesis: ( IC in dom p implies DecIC (p,k) = (DataPart p) +* (Start-At (((IC p) -' k),S)) )
A1: dom (Start-At (((IC p) -' k),S)) = {(IC )}
.= dom (Start-At ((IC p),S)) ;
assume A2: IC in dom p ; :: thesis: DecIC (p,k) = (DataPart p) +* (Start-At (((IC p) -' k),S))
thus DecIC (p,k) = ((DataPart p) +* (Start-At ((IC p),S))) +* (Start-At (((IC p) -' k),S)) by A2, Th26
.= (DataPart p) +* (Start-At (((IC p) -' k),S)) by A1, FUNCT_4:74 ; :: thesis: verum