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<b>let </b><font color="Maroon">X</font> be   <a href="csspace.html#NM1">ComplexUnitarySpace</a>;<br/><b>let </b><font color="Maroon">seq1</font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">seq2</font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">seq3</font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">X</font>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:77"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">seq1</font> <a href="clvect_2.html#R2" target="_self">is_compared_to</a> <font color="Maroon">seq2</font>
 <b>and </b><br/><a NAME="E2:77"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">seq2</font> <a href="clvect_2.html#R2" target="_self">is_compared_to</a> <font color="Maroon">seq3</font>
 ;<br/>

<b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="clvect_2.html#D9" target="_self">CLVECT_2:def 9</a><br/>

<b>assume </b><a NAME="E3:77"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/>

<a NAME="E4:77"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="seq_2.html#T3">SEQ_2:3</a></i>;<br/>
<b>then consider </b><font color="Maroon">m1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:77"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m1</font> holds <br/> <a href="csspace.html#K15">dist</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="ref" href="clvect_2.html#T1" target="_self">Th1</a>, </i>;<br/>
<b>consider </b><font color="Maroon">m2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E8:77"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m2</font> holds <br/> <a href="csspace.html#K15">dist</a> <span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">seq3</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i><a class="txt" href="clvect_2.html#E1:77"><i><font color="Green">E43</font></i></a>, <a class="txt" href="clvect_2.html#E2:77"><i><font color="Green">E44</font></i></a>, </i>;<br/>
<b>take </b><font color="Maroon">m</font> = <font color="Maroon">m1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m2</font>;<br/>

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E9:77"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m</font>
 ;<br/>

<a NAME="E10:77"/>
<font color="Maroon">m1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m1</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E11:77"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m1</font>
 
<b>by </b><i><a class="txt" href="clvect_2.html#E6:77"><i><font color="Green">E47</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E12:77"/><b>then </b><i><font color="Green">E52</font></i>: 
 <a href="csspace.html#K15">dist</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="ref" href="clvect_2.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E13:77"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m2</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E14:77"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m2</font>
 
<b>by </b><i><a class="txt" href="clvect_2.html#E6:77"><i><font color="Green">E47</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E15:77"/><b>then </b>
 <a href="csspace.html#K15">dist</a> <span class="p1">(<span class="default"><font color="Maroon">seq2</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">seq3</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="txt" href="clvect_2.html#E4:77"><i><font color="Green">E46</font></i></a></i>;<br/>
<a NAME="E16:77"/><b>then </b><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default"><a href="csspace.html#K15">dist</a> <span class="p2">(<span class="default"><font color="Maroon">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">seq2</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="csspace.html#K15">dist</a> <span class="p2">(<span class="default"><font color="Maroon">seq2</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">seq3</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E17:77"/>
 <a href="csspace.html#K15">dist</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">seq3</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="csspace.html#K15">dist</a> <span class="p2">(<span class="default"><font color="Maroon">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">seq2</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="csspace.html#K15">dist</a> <span class="p2">(<span class="default"><font color="Maroon">seq2</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">seq3</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="csspace.html#T54">CSSPACE:54</a></i>;<br/>
<b>hence </b><a NAME="E18:77"/>
 <a href="csspace.html#K15">dist</a> <span class="p1">(<span class="default"><font color="Maroon">seq1</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">seq3</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


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