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<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="bhsp_1.html#NM1" title="BHSP_1:NM.1">RealUnitarySpace</a>;<br/><b>let </b><font color="Maroon" title="c2">g</font>, <font color="Maroon" title="c3">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c4">seq</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">X</font>;<br/>







<b>assume </b><a NAME="E1:40"/>
( <font color="Maroon" title="c4">seq</font> is <a href="bhsp_2.html#V1" title="BHSP_2:attr.1">convergent</a> &amp;  <a href="bhsp_2.html#K1" title="BHSP_2:func.1">lim</a> <font color="Maroon" title="c4">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> )
 ;<br/>

<a NAME="E2:40"/><b>then </b>
( <span class="p1"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">seq</font> <a href="bhsp_1.html#K7" title="BHSP_1:func.7">+</a> <span class="p4">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p4">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span></span>)</span></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">seq</font> <a href="bhsp_1.html#K7" title="BHSP_1:func.7">+</a> <span class="p4">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p4">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span></span>)</span></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="bhsp_2.html#T33" target="_self" title="BHSP_2:th.33">Th33</a></i>;<br/>
<a NAME="E3:40"/><b>then </b>
( <span class="p1"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">seq</font> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p4">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span></span>)</span></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">seq</font> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p4">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span></span>)</span></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="bhsp_1.html#T78" title="BHSP_1:th.78">BHSP_1:78</a></i>;<br/>
<b>hence </b><a NAME="E4:40"/>
( <span class="p1"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">seq</font> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">seq</font> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="normsp_1.html#K5" title="NORMSP_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="rlvect_1.html#K5" title="RLVECT_1:func.5">-</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span></span><a href="bhsp_2.html#K2" title="BHSP_2:func.2">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 <b>by </b><i><a class="ref" href="rlvect_1.html#D11" title="RLVECT_1:def.11">RLVECT_1:def 11</a></i>;<br/>


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