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 (R1) (R3) (R6) implies (a")" * b <<>> a * b
  proof
    assume
A1: S is (R1) (R3) (R6);
    take (a""*1.S)*b;
A2: (a""*1.S)*b =*=> a""*(1.S*b) by A1,Th2;
    1.S*b ==> b by A1; then
    a""*(1.S*b) =*=> a""*b by Th2,ThI3;
    hence (a""*1.S)*b =*=> a""*b by A2,Th3;
    a"" * 1.S ==> a by A1;
    hence (a"" * 1.S) * b =*=> a * b by Th2,ThI2;
  end;
