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 (R4) (R8) (R13) implies a * (b * a)" <<>> b"
  proof
    assume
A1: S is (R4) (R8) (R13);
    take b"*(b*(a*(b*a)"));
    thus b"*(b*(a*(b*a)")) =*=> a*(b*a)" by A1,Th2;
    b"*(b*(a*(b*a)")) ==> b"*1.S & b"*1.S ==> b" by A1,ThI3;
    hence b"*(b*(a*(b*a)")) =*=> b" by Lem3;
  end;
