"properly included" meaning in English

{
  "antonyms": [
    {
      "word": "improperly included"
    },
    {
      "word": "equal to"
    }
  ],
  "head_templates": [
    {
      "args": {
        "1": "-",
        "head": "properly included"
      },
      "expansion": "properly included (not comparable)",
      "name": "en-adj"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "adj",
  "senses": [
    {
      "categories": [
        {
          "kind": "other",
          "name": "English entries with incorrect language header",
          "parents": [],
          "source": "w"
        },
        {
          "kind": "other",
          "name": "Pages with 1 entry",
          "parents": [],
          "source": "w"
        },
        {
          "kind": "other",
          "name": "Pages with entries",
          "parents": [],
          "source": "w"
        },
        {
          "kind": "other",
          "langcode": "en",
          "name": "Set theory",
          "orig": "en:Set theory",
          "parents": [],
          "source": "w"
        }
      ],
      "examples": [
        {
          "bold_text_offsets": [
            [
              54,
              71
            ]
          ],
          "ref": "2003, Herbert B. Enderton, Elements of Set Theory, Academic Press, page 22:",
          "text": "If A is a subset of B but A ≠ B, then A is said to be properly included in B.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "Being a proper subset; having the property that every element of a set A is also an element of a set B, and A is not equal to B."
      ],
      "id": "en-properly_included-en-adj-5zKBft~d",
      "links": [
        [
          "set theory",
          "set theory"
        ],
        [
          "proper subset",
          "proper subset"
        ]
      ],
      "raw_glosses": [
        "(set theory) Being a proper subset; having the property that every element of a set A is also an element of a set B, and A is not equal to B."
      ],
      "related": [
        {
          "word": "proper inclusion"
        }
      ],
      "synonyms": [
        {
          "word": "proper subset"
        }
      ],
      "tags": [
        "not-comparable"
      ],
      "topics": [
        "mathematics",
        "sciences",
        "set-theory"
      ]
    }
  ],
  "word": "properly included"
}

{
  "antonyms": [
    {
      "word": "improperly included"
    },
    {
      "word": "equal to"
    }
  ],
  "head_templates": [
    {
      "args": {
        "1": "-",
        "head": "properly included"
      },
      "expansion": "properly included (not comparable)",
      "name": "en-adj"
    }
  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "adj",
  "related": [
    {
      "word": "proper inclusion"
    }
  ],
  "senses": [
    {
      "categories": [
        "English adjectives",
        "English entries with incorrect language header",
        "English lemmas",
        "English multiword terms",
        "English terms with quotations",
        "English uncomparable adjectives",
        "Pages with 1 entry",
        "Pages with entries",
        "en:Set theory"
      ],
      "examples": [
        {
          "bold_text_offsets": [
            [
              54,
              71
            ]
          ],
          "ref": "2003, Herbert B. Enderton, Elements of Set Theory, Academic Press, page 22:",
          "text": "If A is a subset of B but A ≠ B, then A is said to be properly included in B.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "Being a proper subset; having the property that every element of a set A is also an element of a set B, and A is not equal to B."
      ],
      "links": [
        [
          "set theory",
          "set theory"
        ],
        [
          "proper subset",
          "proper subset"
        ]
      ],
      "raw_glosses": [
        "(set theory) Being a proper subset; having the property that every element of a set A is also an element of a set B, and A is not equal to B."
      ],
      "tags": [
        "not-comparable"
      ],
      "topics": [
        "mathematics",
        "sciences",
        "set-theory"
      ]
    }
  ],
  "synonyms": [
    {
      "word": "proper subset"
    }
  ],
  "word": "properly included"
}