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Mr Cull acknowledged Mr Avery's 14 years of service to the council in a media statement issued late yesterday.
His comments came after Mr Avery on Thursday announced his resignation, saying the ''buck stops with me'' over the alleged fraud.
His decision to resign, despite a Deloitte investigation finding he had no knowledge of the alleged fraud, showed he was an ''honourable'' public servant, council chief executive Sue Bidrose said at the time.
Mr Cull said yesterday he also wanted to express his ''personal acknowledgement'' of Mr Avery's decision and his service to the city.
''He [Mr Avery] is someone I have always found to act with integrity and his decision to resign in this way and say that the buck stops with him is just another example of that.
''Tony's experience and wide range of skills means he will be very difficult to replace, but it shows the integrity of the man that he is willing to stay on while we recruit for his replacement.''
Mr Cull also expressed his ''complete support'' for the work of Dr Bidrose in addressing ''organisational issues'' within the council.
''We tasked Sue and past CEO Paul Orders with reviewing DCC practices with a view to identifying and fixing any issues that arose to ensure our accountability to ratepayers.
''That work is ongoing.''