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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   <a href="bhsp_1.html#NM1" title="BHSP_1:NM.1">RealUnitarySpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">g</font>, <font color="Olive" title="b2">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">X</font><br/> for <font color="Olive" title="b3">seq</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b3">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="Olive" title="b3">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> holds <br/>( <span class="p1"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">seq</font> <a href="bhsp_1.html#K6" title="BHSP_1:func.6">+</a> <font color="Olive" title="b2">x</font></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"><font color="Olive" title="b3">seq</font> <a href="bhsp_1.html#K6" title="BHSP_1:func.6">+</a> <font color="Olive" title="b2">x</font></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> <span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">g</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">x</font></span>)</span></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> )</span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">seq</font> being  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">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="Olive" title="b1">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> holds <br/>( <span class="p1"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">seq</font> <a href="bhsp_1.html#K6" title="BHSP_1:func.6">+</a> <font color="Maroon" title="c3">x</font></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"><font color="Olive" title="b1">seq</font> <a href="bhsp_1.html#K6" title="BHSP_1:func.6">+</a> <font color="Maroon" title="c3">x</font></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> <span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">x</font></span>)</span></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> )</span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <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> implies ( <span class="p1"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">seq</font> <a href="bhsp_1.html#K6" title="BHSP_1:func.6">+</a> <font color="Maroon" title="c3">x</font></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"><font color="Maroon" title="c4">seq</font> <a href="bhsp_1.html#K6" title="BHSP_1:func.6">+</a> <font color="Maroon" title="c3">x</font></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> <span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">x</font></span>)</span></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> ) )</span><br/>







<span class="kw">assume </span><a NAME="E1:37"/>
( <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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">seq</font> <a href="bhsp_1.html#K6" title="BHSP_1:func.6">+</a> <font color="Maroon" title="c3">x</font></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"><font color="Maroon" title="c4">seq</font> <a href="bhsp_1.html#K6" title="BHSP_1:func.6">+</a> <font color="Maroon" title="c3">x</font></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> <span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">x</font></span>)</span></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> )</span><br/>

<a NAME="E2:37"/><span class="kw">then </span>
( <font color="Maroon" title="c4">seq</font> <a href="bhsp_1.html#K6" title="BHSP_1:func.6">+</a> <font color="Maroon" title="c3">x</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> <span class="p1">(<span class="default"><font color="Maroon" title="c4">seq</font> <a href="bhsp_1.html#K6" title="BHSP_1:func.6">+</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">x</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bhsp_2.html#T7" target="_self" title="BHSP_2:th.7">Th7</a>, <a class="ref" href="bhsp_2.html#T17" target="_self" title="BHSP_2:th.17">Th17</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:37"/>
( <span class="p1"><a href="bhsp_2.html#K2" title="BHSP_2:func.2">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">seq</font> <a href="bhsp_1.html#K6" title="BHSP_1:func.6">+</a> <font color="Maroon" title="c3">x</font></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"><font color="Maroon" title="c4">seq</font> <a href="bhsp_1.html#K6" title="BHSP_1:func.6">+</a> <font color="Maroon" title="c3">x</font></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> <span class="p1"><a href="bhsp_1.html#K3" title="BHSP_1:func.3">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c3">x</font></span>)</span></span><a href="bhsp_1.html#K3" title="BHSP_1:func.3">.||</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="bhsp_2.html#T21" target="_self" title="BHSP_2:th.21">Th21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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