theorem
  F is having_a_unity & F is associative & F is having_an_inverseOp
  implies F.:(T,(the_inverseOp_wrt F)*T) = i|->the_unity_wrt F &
  F.:((the_inverseOp_wrt F)*T,T) = i|->the_unity_wrt F
proof
  assume
A1: F is having_a_unity & F is associative & F is having_an_inverseOp;
  reconsider uT = the_inverseOp_wrt F*T as Element of i-tuples_on D by
FINSEQ_2:113;
  per cases;
  suppose
A2: i = 0;
    then F.:(T,uT) = <*>D & F.:(uT,T) = <*>D by Lm1;
    hence thesis by A2;
  end;
  suppose
    i <> 0;
    then reconsider C = Seg i as non empty set;
    T is Function of C,D by Lm4;
    hence thesis by A1,Th71;
  end;
end;
