How Typescript Intersection and Omit works with generic types

Question

Code

type Props = {
  prop: string;
};

export default function withProps<T extends Props>(
  Component: React.ComponentType<T>
) {
  
  function WithProps({ prop, ...rest }: T) {
    // Error: Type '{ prop: string; } & Omit<T, "prop">' is not assignable to type 'T'.
    const props: T = {
      prop,
      ...rest
    };
    return <Component {...props} />;
  }

  return (props: T) => <WithProps prop="someProp" {...props} />;
}

The code is just an intended demo to demonstrate my question

Error

In line 11, Typescript complains that Type '{ prop: string; } & Omit<T, "prop">' is not assignable to type 'T', which is weird to me since I think '{ prop: string; } & Omit<T, "prop">' should be identical to T when T extends Props.

My question is why is this seen as an error by TS and how to fix it.