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 k being Element of NAT
for p being PartState of S st IC S in dom p holds
IncrIC ((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 k being Element of NAT
for p being PartState of S st IC S in dom p holds
IncrIC ((NPP p),k) = (DataPart p) +* (Start-At (((IC p) + k),S))
let k be Element of NAT ; :: thesis: for p being PartState of S st IC S in dom p holds
IncrIC ((NPP p),k) = (DataPart p) +* (Start-At (((IC p) + k),S))
let p be PartState of S; :: thesis: ( IC S in dom p implies IncrIC ((NPP p),k) = (DataPart p) +* (Start-At (((IC p) + k),S)) )
A: dom (Start-At (((IC p) + k),S)) = {(IC S)} by FUNCOP_1:19
.= dom (Start-At ((IC p),S)) by FUNCOP_1:19 ;
assume Z: IC S in dom p ; :: thesis: IncrIC ((NPP p),k) = (DataPart p) +* (Start-At (((IC p) + k),S))
hence IncrIC ((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 Z, Th74
.= (DataPart p) +* (Start-At (((IC p) + k),S)) by A, FUNCT_4:78 ;
:: thesis: verum