Apparatus for use at a native valve of a heart of a subject, the apparatus comprising: a delivery tool, comprising: a shaft, having a shaft-axis; a capsule, disposed at a distal portion of the tool, and comprising an upstream capsule-portion and a downstream capsule-portion, the capsule being openable by moving the upstream capsule-portion and the downstream capsule-portion apart; and a balloon, coupled to the shaft, and disposed within the capsule; and a prosthetic valve, comprising: a tubular frame that circumscribes a longitudinal axis to define a lumen along the longitudinal axis, that is compressed around the balloon, and that is disposed within the capsule; and an outer frame coupled to the tubular frame, the outer frame comprising: one or more shape-memory flanges, constrained within the downstream capsule-portion configured to engage a downstream surface of a native heart valve; and a shape-memory upstream support portion configured to be placed against an upstream surface of a native heart valve, constrained within the upstream capsule-portion, wherein: the flanges are configured to automatically deflect radially outward from the tubular frame upon exposure from the downstream capsule-portion, the upstream support portion is configured to automatically deflect radially outward upon exposure from the upstream capsule-portion, the tubular frame is configured to remain compressed around the balloon upon exposure of the tubular frame from the capsule, the outer frame is disposed radially outward from the tubular frame, and while the tubular frame is exposed from the capsule: the tubular frame is configured to inhibit the outer frame from radially expanding, prior to inflation of the balloon, and inflation of the balloon plastically expands the tubular frame radially. a delivery tool, comprising: a shaft, having a shaft-axis; a capsule, disposed at a distal portion of the tool, and comprising an upstream capsule-portion and a downstream capsule-portion, the capsule being openable by moving the upstream capsule-portion and the downstream capsule-portion apart; and a balloon, coupled to the shaft, and disposed within the capsule; and a shaft, having a shaft-axis; a capsule, disposed at a distal portion of the tool, and comprising an upstream capsule-portion and a downstream capsule-portion, the capsule being openable by moving the upstream capsule-portion and the downstream capsule-portion apart; and a balloon, coupled to the shaft, and disposed within the capsule; and a prosthetic valve, comprising: a tubular frame that circumscribes a longitudinal axis to define a lumen along the longitudinal axis, that is compressed around the balloon, and that is disposed within the capsule; and an outer frame coupled to the tubular frame, the outer frame comprising: one or more shape-memory flanges, constrained within the downstream capsule-portion configured to engage a downstream surface of a native heart valve; and a shape-memory upstream support portion configured to be placed against an upstream surface of a native heart valve, constrained within the upstream capsule-portion, a tubular frame that circumscribes a longitudinal axis to define a lumen along the longitudinal axis, that is compressed around the balloon, and that is disposed within the capsule; and an outer frame coupled to the tubular frame, the outer frame comprising: one or more shape-memory flanges, constrained within the downstream capsule-portion configured to engage a downstream surface of a native heart valve; and a shape-memory upstream support portion configured to be placed against an upstream surface of a native heart valve, constrained within the upstream capsule-portion, one or more shape-memory flanges, constrained within the downstream capsule-portion configured to engage a downstream surface of a native heart valve; and a shape-memory upstream support portion configured to be placed against an upstream surface of a native heart valve, constrained within the upstream capsule-portion, the flanges are configured to automatically deflect radially outward from the tubular frame upon exposure from the downstream capsule-portion, the upstream support portion is configured to automatically deflect radially outward upon exposure from the upstream capsule-portion, the tubular frame is configured to remain compressed around the balloon upon exposure of the tubular frame from the capsule, the outer frame is disposed radially outward from the tubular frame, and while the tubular frame is exposed from the capsule: the tubular frame is configured to inhibit the outer frame from radially expanding, prior to inflation of the balloon, and inflation of the balloon plastically expands the tubular frame radially. the tubular frame is configured to inhibit the outer frame from radially expanding, prior to inflation of the balloon, and inflation of the balloon plastically expands the tubular frame radially.