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 (R12) (R15) implies a * (b * a)" =*=> b"
  proof
    assume
A1: S is (R12) (R15);
    a * (b * a)" ==> a*(a"*b") & a*(a"*b") ==> b" by A1,ThI3;
    hence a * (b * a)" =*=> b" by Lem3;
  end;
