The young waiter was talking to someone at the bar when she came in, and at first he did not pay any attention to her, because he had not looked at her properly. Then he turned round and saw how beautiful she was. His face went bright red, and he hurried over to take her order.
I looked up the dictionaries, "properly" can have the meaning of "really, or completely". So does the sentence in bold mean the waiter didn't look at the girl completely?
. he had not looked at her carefully, in the way that he should have looked at her.
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. . . he had not looked at her carefully, in the way that he should have looked at her.