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
for k being Nat st IC in dom p holds
DecIC ((NPP p),k) = (DataPart p) +* (Start-At (((IC p) -' k),S))
let S be non empty stored-program IC-Ins-separated definite realistic COM-Struct of N; :: thesis: for p being PartState of S
for k being Nat st IC in dom p holds
DecIC ((NPP 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 ((NPP p),k) = (DataPart p) +* (Start-At (((IC p) -' k),S))
let k be Nat; :: thesis: ( IC in dom p implies DecIC ((NPP p),k) = (DataPart p) +* (Start-At (((IC p) -' k),S)) )
A1: dom (Start-At (((IC p) -' k),S)) = {(IC )} by FUNCOP_1:19
.= dom (Start-At ((IC p),S)) by FUNCOP_1:19 ;
assume A2: IC in dom p ; :: thesis: DecIC ((NPP p),k) = (DataPart p) +* (Start-At (((IC p) -' k),S))
hence DecIC ((NPP p),k) = (NPP p) +* (Start-At (((IC p) -' k),S)) by Th72
.= ((DataPart p) +* (Start-At ((IC p),S))) +* (Start-At (((IC p) -' k),S)) by A2, Th74
.= (DataPart p) +* (Start-At (((IC p) -' k),S)) by A1, FUNCT_4:78 ;
:: thesis: verum