reserve X for ARS, a,b,c,u,v,w,x,y,z for Element of X;
reserve i,j,k for Element of ARS_01;
reserve l,m,n for Element of ARS_02;
reserve A for set;

theorem
  for X being non empty ARS holds X is COMP iff the reduction of X is complete
  proof let X be non empty ARS;
    set R = the reduction of X;
A2: X is CONF iff R is confluent by Ch17;
    X is SN iff R is strongly-normalizing by Ch7,Ch8;
    hence thesis by A2;
  end;
