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 x being set st x in dom Initialize p holds x in dom p or x = IC S
proof
  let x be set;
  assume
A1: x in dom Initialize p;
  dom Initialize p = dom p \/ {IC S} by Th42;
  then x in dom p or x in {IC S} by A1,XBOOLE_0:def 3;
  hence thesis by TARSKI:def 1;
end;
