let N be with_non-empty_elements set ; :: thesis: for S being non empty IC-Ins-separated Mem-Struct of N
for p, q being PartState of S
for k being Nat st IC in dom q holds
DecIC ((p +* (IncIC (q,k))),k) = p +* q
let S be non empty IC-Ins-separated Mem-Struct of N; :: thesis: for p, q being PartState of S
for k being Nat st IC in dom q holds
DecIC ((p +* (IncIC (q,k))),k) = p +* q
let p, q be PartState of S; :: thesis: for k being Nat st IC in dom q holds
DecIC ((p +* (IncIC (q,k))),k) = p +* q
let k be Nat; :: thesis: ( IC in dom q implies DecIC ((p +* (IncIC (q,k))),k) = p +* q )
assume Z: IC in dom q ; :: thesis: DecIC ((p +* (IncIC (q,k))),k) = p +* q
IC in dom (IncIC (q,k)) by Th53;
hence DecIC ((p +* (IncIC (q,k))),k) = p +* (DecIC ((IncIC (q,k)),k)) by Th198
.= p +* q by Z, Th201 ;
:: thesis: verum