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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><a NAME="E1:57"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid_4.html#K4" target="_self">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K8">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="euclid_4.html#K4" target="_self">.|</a></span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="euclid_4.html#K2" target="_self">Line</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> </span>}</span> 
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:57"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="euclid_4.html#K2" target="_self">Line</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E4:57"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>6</sub></font> <a href="euclid_4.html#R1" target="_self">are_orthogonal</a> 
 <b>and </b><br/><a NAME="E5:57"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="euclid_4.html#K2" target="_self">Line</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> holds <br/><span class="p1"><a href="euclid_4.html#K4" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="euclid_4.html#K4" target="_self">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="euclid_4.html#K4" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K8">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="euclid_4.html#K4" target="_self">.|</a></span>
 <b>by </b><i><a class="txt" href="euclid_4.html#E63"><i><font color="Green">Lemma40</font></i></a></i>;<br/>
<a NAME="E6:57"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="euclid_4.html#K2" target="_self">Line</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="euclid_4.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E7:57"/><b>then </b><i><font color="Green">E45</font></i>: 
<span class="p1"><a href="euclid_4.html#K4" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid_4.html#K4" target="_self">.|</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="euclid_4.html#E1:57"><i><font color="Green">E41</font></i></a></i>;<br/>
<a NAME="E8:57"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="euclid_4.html#K4" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="euclid_4.html#K4" target="_self">.|</a></span> <a href="xcmplx_0.html#K2">+</a> <font color="Olive">b<sub>1</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:57_1"/><i><font color="Green">E47</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<b>take </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1"><a href="euclid_4.html#K4" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="euclid_4.html#K4" target="_self">.|</a></span>;<br/>

<b>thus </b><a NAME="E2:57_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="euclid_4.html#K4" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="euclid_4.html#K4" target="_self">.|</a></span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="euclid_4.html#E1:57"><i><font color="Green">E41</font></i></a>, <a class="txt" href="euclid_4.html#E3:57"><i><font color="Green">E42</font></i></a>, <a class="txt" href="euclid_4.html#E1:57_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:57"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><span class="p1"><a href="euclid_4.html#K4" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="euclid_4.html#K4" target="_self">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>assume </b><a NAME="E1:57_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:57_2"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid_4.html#K4" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span><a href="euclid_4.html#K4" target="_self">.|</a></span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="euclid_4.html#K2" target="_self">Line</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="euclid_4.html#E1:57"><i><font color="Green">E41</font></i></a></i>;<br/>
<b>thus </b><a NAME="E4:57_2"/>
<span class="p1"><a href="euclid_4.html#K4" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="euclid_4.html#K4" target="_self">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="euclid_4.html#E5:57"><i><font color="Green">E44</font></i></a>, <a class="txt" href="euclid_4.html#E3:57_2"><i><font color="Green">E48</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:57"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> is <a href="seq_4.html#V2">bounded_below</a>
 
<b>by </b><i><a class="ref" href="seq_4.html#D2">SEQ_4:def 2</a></i>;<br/>
<a NAME="E11:57"/><b>then </b>
<span class="p1"><a href="euclid_4.html#K4" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span><a href="euclid_4.html#K4" target="_self">.|</a></span> <a href="hidden.html#R1">=</a>  <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="euclid_4.html#E7:57"><i><font color="Green">E45</font></i></a>, <a class="txt" href="euclid_4.html#E8:57"><i><font color="Green">E46</font></i></a>, <a class="txt" href="euclid_4.html#E9:57"><i><font color="Green">E47</font></i></a>, <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/>
<b>hence </b><a NAME="E12:57"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="euclid_4.html#K2" target="_self">Line</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp; <span class="p1"><a href="euclid_4.html#K4" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K8">-</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="euclid_4.html#K4" target="_self">.|</a></span> <a href="hidden.html#R1">=</a>  <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K8">-</a> <font color="Olive">b<sub>1</sub></font> <a href="euclid_4.html#R1" target="_self">are_orthogonal</a>  )
 <b>by </b><i><a class="txt" href="euclid_4.html#E3:57"><i><font color="Green">E42</font></i></a>, <a class="txt" href="euclid_4.html#E4:57"><i><font color="Green">E43</font></i></a></i>;<br/>


</div>
