Cfrgtkky

documentclass{article}





usepackage{newtxtext}

usepackage{newtxmath}

usepackage{amsmath}





begin{document}



It is clearer if you define $q_i = p_i cdot f_i cdot overline{f}_i$



This is how it should be.



Then



$p cdot f = p cdot b_i cdot f_i qquad text{(since } f = b_i cdot f_i)$



$= d^*_i cdot p_i cdot f_i qquadtext{(by naturality)}$



$= d^*_i cdot p_i cdot f_i cdot overline{f}_i cdot f_i$ = qquad (as $overline{f}_i cdot f_i = 1$)



$= d^*_i cdot q_i cdot f_i  qquad text{(by the definition of} q_i)$



$= q cdot b_i cdot f_i  qquad text{(by naturality)}$



$= q cdot f  text{(by naturality)}$



Hope that helps.



end{document}

documentclass{article}





usepackage{newtxtext}

usepackage{newtxmath}

usepackage{amsmath}





begin{document}



It is clearer if you define $q_i = p_i cdot f_i cdot overline{f}_i$



This is how it should be.



Then



$p cdot f = p cdot b_i cdot f_i qquad text{(since } f = b_i cdot f_i)$



$= d^*_i cdot p_i cdot f_i qquadtext{(by naturality)}$



$= d^*_i cdot p_i cdot f_i cdot overline{f}_i cdot f_i$ = qquad (as $overline{f}_i cdot f_i = 1$)



$= d^*_i cdot q_i cdot f_i  qquad text{(by the definition of} q_i)$



$= q cdot b_i cdot f_i  qquad text{(by naturality)}$



$= q cdot f  text{(by naturality)}$



Hope that helps.



end{document}

documentclass{article}





usepackage{newtxtext}

usepackage{newtxmath}

usepackage{amsmath}





begin{document}



It is clearer if you define $q_i = p_i cdot f_i cdot overline{f}_i$



This is how it should be.



Then



$p cdot f = p cdot b_i cdot f_i qquad text{(since } f = b_i cdot f_i)$



$= d^*_i cdot p_i cdot f_i qquadtext{(by naturality)}$



$= d^*_i cdot p_i cdot f_i cdot overline{f}_i cdot f_i$ = qquad (as $overline{f}_i cdot f_i = 1$)



$= d^*_i cdot q_i cdot f_i  qquad text{(by the definition of} q_i)$



$= q cdot b_i cdot f_i  qquad text{(by naturality)}$



$= q cdot f  text{(by naturality)}$



Hope that helps.



end{document}

documentclass{article}





usepackage{newtxtext}

usepackage{newtxmath}

usepackage{amsmath}





begin{document}



It is clearer if you define $q_i = p_i cdot f_i cdot overline{f}_i$



This is how it should be.



Then



$p cdot f = p cdot b_i cdot f_i qquad text{(since } f = b_i cdot f_i)$



$= d^*_i cdot p_i cdot f_i qquadtext{(by naturality)}$



$= d^*_i cdot p_i cdot f_i cdot overline{f}_i cdot f_i$ = qquad (as $overline{f}_i cdot f_i = 1$)



$= d^*_i cdot q_i cdot f_i  qquad text{(by the definition of} q_i)$



$= q cdot b_i cdot f_i  qquad text{(by naturality)}$



$= q cdot f  text{(by naturality)}$



Hope that helps.



end{document}

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documentclass{article}

usepackage{amsmath}



begin{document}



begin{align}

 &text{It is clearer if you define} &q_i &= p_i cdot f_i cdot overline{f}_i\

&text{This is how it should be} & p cdot f &= p cdot b_i cdot f_i\

&text{then} &&= d^*_i cdot p_i cdot f_i qquad text{(by neutrality)}

end{align}

end{document}

documentclass{article}

usepackage{amsmath}



begin{document}



begin{align}

 text{It is clearer if you define}& &q_i &= p_i cdot f_i cdot overline{f}_i\

text{This is how it should be}& & p cdot f &= p cdot b_i cdot f_i\

text{then} &&&= d^*_i cdot p_i cdot f_i qquad text{(by neutrality)}

end{align}

end{document}

documentclass{article}

usepackage{amsmath}



begin{document}



begin{align}

 &text{It is clearer if you define} &q_i &= p_i cdot f_i cdot overline{f}_i\

&text{This is how it should be} & p cdot f &= p cdot b_i cdot f_i\

&text{then} &&= d^*_i cdot p_i cdot f_i qquad text{(by neutrality)}

end{align}

end{document}

documentclass{article}

usepackage{amsmath}



begin{document}



begin{align}

 text{It is clearer if you define}& &q_i &= p_i cdot f_i cdot overline{f}_i\

text{This is how it should be}& & p cdot f &= p cdot b_i cdot f_i\

text{then} &&&= d^*_i cdot p_i cdot f_i qquad text{(by neutrality)}

end{align}

end{document}

documentclass{article}

usepackage{amsmath}



begin{document}



begin{align}

 &text{It is clearer if you define} &q_i &= p_i cdot f_i cdot overline{f}_i\

&text{This is how it should be} & p cdot f &= p cdot b_i cdot f_i\

&text{then} &&= d^*_i cdot p_i cdot f_i qquad text{(by neutrality)}

end{align}

end{document}

documentclass{article}

usepackage{amsmath}



begin{document}



begin{align}

 text{It is clearer if you define}& &q_i &= p_i cdot f_i cdot overline{f}_i\

text{This is how it should be}& & p cdot f &= p cdot b_i cdot f_i\

text{then} &&&= d^*_i cdot p_i cdot f_i qquad text{(by neutrality)}

end{align}

end{document}

documentclass{article}

usepackage{amsmath}



begin{document}



begin{align}

 &text{It is clearer if you define} &q_i &= p_i cdot f_i cdot overline{f}_i\

&text{This is how it should be} & p cdot f &= p cdot b_i cdot f_i\

&text{then} &&= d^*_i cdot p_i cdot f_i qquad text{(by neutrality)}

end{align}

end{document}

documentclass{article}

usepackage{amsmath}



begin{document}



begin{align}

 text{It is clearer if you define}& &q_i &= p_i cdot f_i cdot overline{f}_i\

text{This is how it should be}& & p cdot f &= p cdot b_i cdot f_i\

text{then} &&&= d^*_i cdot p_i cdot f_i qquad text{(by neutrality)}

end{align}

end{document}

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