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 Th42:
  for S being IC-Ins-separated non empty
  with_non-empty_values Mem-Struct over N,
      p being PartState of S
  holds dom Initialize p = dom p \/ {IC S}
proof let S be IC-Ins-separated non empty
   with_non-empty_values Mem-Struct over N;
  let p being PartState of S;
  thus dom Initialize p
     = dom p \/ dom Start-At(0,S) by FUNCT_4:def 1
    .= dom p \/ {IC S};
end;
