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<b>let </b><font color="Maroon">X</font> be   <a href="bhsp_1.html#NM1">RealUnitarySpace</a>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">seq</font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">X</font>;<br/>







<b>assume </b><a NAME="E1:40"/>
( <font color="Maroon">seq</font> is <a href="bhsp_2.html#V1" target="_self">convergent</a> &amp;  <a href="bhsp_2.html#K1" target="_self">lim</a> <font color="Maroon">seq</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> )
 ;<br/>

<a NAME="E2:40"/><b>then </b>
( <span class="p1"><a href="bhsp_2.html#K2" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="bhsp_1.html#K7">+</a> <span class="p4">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="normsp_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="rlvect_1.html#K4">+</a> <span class="p4">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span></span><a href="bhsp_2.html#K2" target="_self">.||</a></span> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1"><a href="bhsp_2.html#K2" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="bhsp_1.html#K7">+</a> <span class="p4">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span></span>)</span> <a href="normsp_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="rlvect_1.html#K4">+</a> <span class="p4">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span></span><a href="bhsp_2.html#K2" target="_self">.||</a></span> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i/>;<br/>
<a NAME="E3:40"/><b>then </b>
( <span class="p1"><a href="bhsp_2.html#K2" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="normsp_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="rlvect_1.html#K4">+</a> <span class="p4">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span></span><a href="bhsp_2.html#K2" target="_self">.||</a></span> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1"><a href="bhsp_2.html#K2" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="normsp_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="rlvect_1.html#K4">+</a> <span class="p4">(<span class="default"><a href="rlvect_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span></span>)</span></span>)</span></span><a href="bhsp_2.html#K2" target="_self">.||</a></span> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="bhsp_1.html#T78">BHSP_1:78</a></i>;<br/>
<b>hence </b><a NAME="E4:40"/>
( <span class="p1"><a href="bhsp_2.html#K2" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="normsp_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span></span>)</span></span><a href="bhsp_2.html#K2" target="_self">.||</a></span> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1"><a href="bhsp_2.html#K2" target="_self">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="normsp_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">g</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font></span>)</span></span>)</span></span><a href="bhsp_2.html#K2" target="_self">.||</a></span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="rlvect_1.html#D11">RLVECT_1:def 11</a></i>;<br/>


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