theorem
  F is associative & F is having_an_inverseOp & F is having_a_unity &
  F.:(T1,T2) = i|->the_unity_wrt F implies
    T1 = (the_inverseOp_wrt F)*T2 & (the_inverseOp_wrt F)*T1 = T2
proof
  assume
A1: F is associative & F is having_an_inverseOp & F is having_a_unity &
  F .:(T1,T2) = i|->the_unity_wrt F;
  per cases;
  suppose
    i = 0;
    then T1 = <*>D & T2 = <*>D;
    hence thesis;
  end;
  suppose
    i <> 0;
    then reconsider C = Seg i as non empty set;
    T1 is Function of C,D & T2 is Function of C,D by Lm4;
    hence thesis by A1,Th72;
  end;
end;
