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 j, k being Nat holds DecIC ((p +* (Start-At (j,S))),k) = p +* (Start-At ((j -' 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 j, k being Nat holds DecIC ((p +* (Start-At (j,S))),k) = p +* (Start-At ((j -' k),S))
let p be PartState of S; :: thesis: for j, k being Nat holds DecIC ((p +* (Start-At (j,S))),k) = p +* (Start-At ((j -' k),S))
let j, k be Nat; :: thesis: DecIC ((p +* (Start-At (j,S))),k) = p +* (Start-At ((j -' k),S))
thus DecIC ((p +* (Start-At (j,S))),k) = p +* (Start-At (((IC (p +* (Start-At (j,S)))) -' k),S)) by FUNCT_4:114
.= p +* (Start-At ((j -' k),S)) by Th16 ; :: thesis: verum