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;
reserve
  S for Group-like quasi_total partial invariant non empty non-empty TRSStr;
reserve a,b,c for Element of S;
reserve S for Group-like quasi_total partial non empty non-empty TRSStr;
reserve a,b,c for Element of S;
reserve
  S for Group-like quasi_total partial invariant non empty non-empty TRSStr,
  a,b,c for Element of S;

theorem
  S is (R2) (R9) implies a * (a") <<>> 1.S
  proof
    assume
A1: S is (R2) (R9);
    take a""*a";
    a"" ==> a by A1;
    hence a""*a" =*=> a*a" by Th2,ThI2;
    thus a""*a" =*=> 1.S by A1,Th2;
  end;
