newcommand{pf}[1]{textbf{textit{Proof.}}quadbegin{minipage}[t][][t]{textwidth}footnotesize #1end{minipage}}

pf{$begin{alignedat}{2}
B_{(Y,d)}(a,r)&=big{x|xin Ywedge d(x,a)<rbig}=Ycapbig{x|d(x,a)<rbig}\
&=Ycapbig{x|xin Xwedge d(x,a)<rbig}=Ycap B_{(X,d)}(a,r)
end{alignedat}$}