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 s being State of S, k being Nat st k <= IC s
 holds IC DecIC(s,k) + k = IC s
proof
 let s be State of S, k be Nat such that
A1: k <= IC s;
 thus IC DecIC(s,k) + k = IC s -' k + k by Th65
    .= IC s by A1,XREAL_1:235;
end;
