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 Th48:
 for p being PartState of S holds IC S in dom Initialize p
proof let p be PartState of S;
A1: dom Initialize p = dom p \/ {IC S} by Th42;
  IC S in {IC S} by TARSKI:def 1;
 hence IC S in dom Initialize p by A1,XBOOLE_0:def 3;
end;
