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
 for k being Nat
 holds IC IncIC(s,k) -' k = IC s
proof
 let k be Nat;
 thus IC IncIC(s,k) -' k = IC s + k -' k by Th53
    .= IC s by NAT_D:34;
end;
