Work done, victor and vanquished arrive back at the family home along the Florida coast at Boca Raton and try to get on with their lives as though nothing unusual has taken place, as if they have returned from one of their regular workouts together on a practice court. They say there will be no gloating, no sulking. "In the end," Venus says, "it's a competition. The best competitor wins. As soon as you walk off the court, the competition's over."
This had not been a normal Sunday. Venus and Serena had become the first sisters to contest the singles final in a high-level tournament since women were first allowed to compete at Wimbledon in 1884. In that year, Maud Watson defeated her sister Lillian, presumably by playing the better strokes while coping with a tight corset and managing not to catch her foot in the hem of her long dress.
The Watsons were quintessentially English, daughters of a clergyman who taught at Harrow and also typical devotees of a sport that has traditionally been the preserve of privileged whites. The Williams sisters are African- Americans whose father, a sharecropper's son from Shreveport, Louisiana, refers to himself as King Richard. He publishes a newsletter in which he explains how he made peace with (then gave guidance to) members of the Bloods gang, who shot at his daughters when they played on park courts in Compton, Los Angeles.
To listen to Richard Williams is to get a sense of what it might have been like to have an audience with Walter Mitty. He is too busy developing business, he says, to spend a lot of time watching Venus and Serena play. "Every year I develop three businesses," he says, "and every business I develop, I expect to produce anywhere from $10m to $15m a year. The girls get really upset with me because I won't go [to matches]. They go, `You the coach'; I say, `No, I'm a manager, too'. As a matter of fact, we're thinking about buying Rockefeller Center for $3.9bn, so I don't have time to even think about tennis no more."
This is understandable, seeing that his other projects apparently include "a lot of work for the Chinese peoples and the Japanese peoples and so on". He has taken up singing, too - "I plan to push Michael Jackson, if I can" - and a book is to be published in August. "Some people have said I was mad anyway, so it's called Method to My Madness."
It is an apt title, because whether or not Richard Williams is kidding himself or kidding the media, his achievement as his daughters' mentor is phenomenal. In next to no time, Venus has won more than $2.5m in official prize money alone, and Serena is heading for her first $1m. And that is aside from the millions they receive from sponsorships and endorsements.
The sheer size and power of Venus and Serena is an indication of how women's professional tennis has risen to a new and pulverising level; a level that would have left previous generations gasping. Surprisingly, the Williams sisters were withheld from junior tournaments, forgoing the customary route to the professional game. Their father said he did not want to risk "burn-out".
Their mother, Oracene, does most of the travelling with the girls. "I'm a mother, I'm a wife, I'm a coach," she says. "And at home I have so many other activities, because I take care of the finances. My husband is an idealist, and he's very creative."
Although clearly one source of the family's strength, Richard Williams does not share their religion. "I'm not a Jehovah Witness," he says. "They are. And I've taught them to make decisions. So whatever decision they make, they would make that decision..."
But Richard Williams is strong on family unity. "My mom taught me that family is the oldest human institution; that it is society's basic unit. Entire civilisations have survived or disappeared depending on whether family life was weak or strong. I taught my kids what my mom taught me.
"Tell you something - when Venus was about four years old, my mom said, `You can't raise two kids like I brought you up in the Forties and Fifties'. I said, `Yes, ma'am'. But I brought them up that way anyway, because they started working at two years old. My wife was really upset with me about that. But we never have a problem with what goes on, because they're looking to help each other all the time."
Along with the power and the tennis skills, there is an air of arrogance and defiance about the Williams sisters. They differ from the majority of single-minded, parent-driven tennis prodigies, emboldened partly by their sense of their own differentness, and partly by their solidarity as a pair. "We are really setting the standards for the future generation," Serena has declared. "Tennis is always going to take a step up now that we've come along."
On one occasion, the younger Williams took the opportunity to educate the assembled media about the etymology of the word "ghetto". "It was a German word," she declared during a press conference. "They took the Jewish people out of their homes because the Germans wanted to be on a pedestal compared to the Jews...
"There was no sanitation area, facilities to use inside or anything. So they named it the ghetto..."
A reporter contradicted her. "That goes back to the Middle Ages. That's not just from the 20th century."
She persisted. "That was World War II."
The reporter pressed on: "Goes back way before that."
Serena would not budge: "You have your information and I have mine."
Last week, on the subject of siblings, Serena was asked whether she was aware that the older one tends to have the upper hand mentally. "I was never aware of siblings having the upper hand mentally," she said. "I think that's just something that's in a book. You can't always trust that."
We can only guess what the late Arthur Ashe would have made of the Williamses. The 1975 Wimbledon champion emerged triumphantly from the racial prejudice he encountered growing up in Virginia and became a sporting icon of the civil rights movement, beloved of his peers. The tennis stadium at the US Open is named in his honour. But the Williams family has prospered in different times. Richard Williams is only too happy to acknowledge that for all the drawbacks to being black in America, hard work can bring rewards.
"I don't think a lot of people understand how good America is until they leave this country," he says."I know when I was young, I used to say, `To hell with America'. But when I got a chance to travel, and when you see what it's like in these other countries... brother, I love America."