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 +* q),k) = p +* (DecIC (q,k))
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 +* q),k) = p +* (DecIC (q,k))
let p, q be PartState of S; :: thesis: for k being Nat st IC in dom q holds
DecIC ((p +* q),k) = p +* (DecIC (q,k))
let k be Nat; :: thesis: ( IC in dom q implies DecIC ((p +* q),k) = p +* (DecIC (q,k)) )
assume IC in dom q ; :: thesis: DecIC ((p +* q),k) = p +* (DecIC (q,k))
then IC (p +* q) = IC q by FUNCT_4:13;
hence DecIC ((p +* q),k) = p +* (DecIC (q,k)) by FUNCT_4:14; :: thesis: verum