reserve x,N for set,
        k for Nat;
reserve N for with_zero set;
reserve S for IC-Ins-separated non empty with_non-empty_values
     Mem-Struct over N;
reserve s for State of S;
reserve p for PartState of S;

theorem Th72:
 for p,q being PartState of S, k being Nat st IC S in dom q
  holds DecIC(p+*q,k) = p +* DecIC(q,k)
proof let p,q be PartState of S, k be Nat;
  assume IC S in dom q;
   then IC(p+*q) = IC q by FUNCT_4:13;
 hence DecIC(p+*q,k) = p +* DecIC(q,k) by FUNCT_4:14;
end;
