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, 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 with_non-empty_values IC-Ins-separated Mem-Struct over 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 A1: IC in dom q ; :: thesis: DecIC ((p +* (IncIC (q,k))),k) = p +* q
IC in dom (IncIC (q,k)) by Th52;
hence DecIC ((p +* (IncIC (q,k))),k) = p +* (DecIC ((IncIC (q,k)),k)) by Th72
.= p +* q by A1, Th73 ;
:: thesis: verum