reserve G for Abelian add-associative right_complementable right_zeroed
  non empty addLoopStr;
reserve GS for non empty addLoopStr;
reserve F for Field;
reserve F for Field,
  n for Nat,
  D for non empty set,
  d for Element of D,
  B for BinOp of D,
  C for UnOp of D;

theorem Th8:
  d is_a_unity_wrt B implies n |-> d is_a_unity_wrt product(B,n)
proof
  assume d is_a_unity_wrt B;
  then
A1: B is having_a_unity & d = the_unity_wrt B by BINOP_1:def 8,SETWISEO:def 2;
  now
    let b be Element of (n-tuples_on D) qua non empty set;
    reconsider b9 = b as Element of n-tuples_on D;
    thus product(B,n).(n |-> d,b) = B.:(n |-> d,b9) by Def1
      .= b by A1,FINSEQOP:56;
    thus product(B,n).(b,n |-> d) = B.:(b9,n |-> d) by Def1
      .= b by A1,FINSEQOP:56;
  end;
  hence thesis by BINOP_1:3;
end;
