Apparatus for use with a native valve of a heart of a subject, the apparatus comprising: an implant comprising: a support frame , having a compressed state, and an expanded state in which the support frame defines an opening therethrough, and is dimensioned to be placed against an upstream surface of the native valve such that the opening is disposed over an orifice defined by the native valve; a flexible sheet ; and a valve frame : having a compressed state, and an expanded state in which the valve frame defines a lumen therethrough, and comprising a valve member disposed within the lumen ; and a catheter for percutaneously delivering the implant in a compressed state thereof to the native valve, with the support frame in the compressed state and the valve frame in the compressed state; characterized in that : the valve frame is coupled to the support frame via the flexible sheet such that when the support frame is in its expanded state, and the valve frame is in its expanded state, at least part of the lumen is disposed within the opening , and the valve frame is not in contact with the support frame ; and the implant is disposed relative to the catheter for implanting transfemorally, with the valve frame closer to a distal end of the catheter than the support frame is to the distal end of the catheter . an implant comprising: a support frame , having a compressed state, and an expanded state in which the support frame defines an opening therethrough, and is dimensioned to be placed against an upstream surface of the native valve such that the opening is disposed over an orifice defined by the native valve; a flexible sheet ; and a valve frame : having a compressed state, and an expanded state in which the valve frame defines a lumen therethrough, and comprising a valve member disposed within the lumen ; and a support frame , having a compressed state, and an expanded state in which the support frame defines an opening therethrough, and is dimensioned to be placed against an upstream surface of the native valve such that the opening is disposed over an orifice defined by the native valve; a flexible sheet ; and a valve frame : having a compressed state, and an expanded state in which the valve frame defines a lumen therethrough, and comprising a valve member disposed within the lumen ; and having a compressed state, and an expanded state in which the valve frame defines a lumen therethrough, and comprising a valve member disposed within the lumen ; and a catheter for percutaneously delivering the implant in a compressed state thereof to the native valve, with the support frame in the compressed state and the valve frame in the compressed state; characterized in that : the valve frame is coupled to the support frame via the flexible sheet such that when the support frame is in its expanded state, and the valve frame is in its expanded state, at least part of the lumen is disposed within the opening , and the valve frame is not in contact with the support frame ; and the implant is disposed relative to the catheter for implanting transfemorally, with the valve frame closer to a distal end of the catheter than the support frame is to the distal end of the catheter . the valve frame is coupled to the support frame via the flexible sheet such that when the support frame is in its expanded state, and the valve frame is in its expanded state, at least part of the lumen is disposed within the opening , and the valve frame is not in contact with the support frame ; and the implant is disposed relative to the catheter for implanting transfemorally, with the valve frame closer to a distal end of the catheter than the support frame is to the distal end of the catheter . <EOS>