<?xml version="1.0"?>
<div class="add">

<b>let </b><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="numbers.html#K1">REAL</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:7_1_2"/><i><font color="Green">E3</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; ex <font color="Olive">b<sub>2</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  st <br/>(  <a href="finsop_1.html#K5">dom</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="finsop_1.html#K5">dom</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="bhsp_5.html#K3" target="_self">PO</a> <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> ) &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K33">addreal</a>  <a href="finsop_1.html#K2">"**"</a> <font color="Olive">b<sub>2</sub></font> ) )
 <b>and </b><br/><a NAME="E2:7_1_2"/><i><font color="Green">E4</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="funct_1.html#V2">one-to-one</a> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; ex <font color="Olive">b<sub>2</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  st <br/>(  <a href="finsop_1.html#K5">dom</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="finsop_1.html#K5">dom</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="bhsp_5.html#K3" target="_self">PO</a> <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> ) &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K33">addreal</a>  <a href="finsop_1.html#K2">"**"</a> <font color="Olive">b<sub>2</sub></font> ) )
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:7_1_2"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>and </b><br/><a NAME="E5:7_1_2"/><i><font color="Green">E6</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E6:7_1_2"/><i><font color="Green">E7</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  st <br/>(  <a href="finsop_1.html#K5">dom</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>6</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="bhsp_5.html#K3" target="_self">PO</a> <font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font> ) &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K33">addreal</a>  <a href="finsop_1.html#K2">"**"</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="bhsp_5.html#E1:7_1_2"><i><font color="Green">E3</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E8:7_1_2"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>7</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>and </b><br/><a NAME="E9:7_1_2"/><i><font color="Green">E9</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E10:7_1_2"/><i><font color="Green">E10</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  st <br/>(  <a href="finsop_1.html#K5">dom</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>7</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="bhsp_5.html#K3" target="_self">PO</a> <font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font> ) &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K33">addreal</a>  <a href="finsop_1.html#K2">"**"</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="bhsp_5.html#E2:7_1_2"><i><font color="Green">E4</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E12:7_1_2"/><i><font color="Green">E11</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E13:7_1_2"/><i><font color="Green">E12</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>8</sub></font> holds <br/><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="bhsp_5.html#K3" target="_self">PO</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E14:7_1_2"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K33">addreal</a>  <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="bhsp_5.html#E6:7_1_2"><i><font color="Green">E7</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E16:7_1_2"/><i><font color="Green">E14</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>7</sub></font>
 <b>and </b><br/><a NAME="E17:7_1_2"/><i><font color="Green">E15</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>9</sub></font> holds <br/><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="bhsp_5.html#K3" target="_self">PO</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E18:7_1_2"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <a href="binop_2.html#K33">addreal</a>  <a href="finsop_1.html#K2">"**"</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="bhsp_5.html#E10:7_1_2"><i><font color="Green">E10</font></i></a></i>;<br/>
<a NAME="E19:7_1_2"/><i><font color="Green">E17</font></i>: 
(  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>7</sub></font> &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="funct_2.html#NM2">Permutation</a> of  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>6</sub></font> st <br/>( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>6</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>6</sub></font> ) )
 
<b>by </b><i><a class="txt" href="bhsp_5.html#E4:7_1_2"><i><font color="Green">E5</font></i></a>, <a class="txt" href="bhsp_5.html#E5:7_1_2"><i><font color="Green">E6</font></i></a>, <a class="txt" href="bhsp_5.html#E8:7_1_2"><i><font color="Green">E8</font></i></a>, <a class="txt" href="bhsp_5.html#E9:7_1_2"><i><font color="Green">E9</font></i></a>, <a class="ref" href="bhsp_5.html#T1" target="_self">Th1</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="funct_2.html#NM2">Permutation</a> of  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>6</sub></font><b> such that </b><br/><a NAME="E21:7_1_2"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>10</sub></font>
 <b>and </b><br/><a NAME="E22:7_1_2"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E23:7_1_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="bhsp_5.html#E4:7_1_2"><i><font color="Green">E5</font></i></a>, <a class="txt" href="bhsp_5.html#E5:7_1_2"><i><font color="Green">E6</font></i></a>, <a class="txt" href="bhsp_5.html#E8:7_1_2"><i><font color="Green">E8</font></i></a>, <a class="txt" href="bhsp_5.html#E9:7_1_2"><i><font color="Green">E9</font></i></a>, <a class="ref" href="bhsp_5.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E24:7_1_2"/><i><font color="Green">E19</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>8</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>8</sub></font> )
 
<b>by </b><i><a class="txt" href="bhsp_5.html#E12:7_1_2"><i><font color="Green">E11</font></i></a>, <a class="ref" href="funct_2.html#T67">FUNCT_2:67</a>, <a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E25:7_1_2"/><i><font color="Green">E20</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>9</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>9</sub></font> )
 
<b>by </b><i><a class="txt" href="bhsp_5.html#E16:7_1_2"><i><font color="Green">E14</font></i></a>, <a class="txt" href="bhsp_5.html#E19:7_1_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="funct_2.html#T67">FUNCT_2:67</a>, <a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<a NAME="E26:7_1_2"/><i><font color="Green">E21</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="bhsp_5.html#E4:7_1_2"><i><font color="Green">E5</font></i></a>, <a class="txt" href="bhsp_5.html#E5:7_1_2"><i><font color="Green">E6</font></i></a>, <a class="txt" href="bhsp_5.html#E8:7_1_2"><i><font color="Green">E8</font></i></a>, <a class="txt" href="bhsp_5.html#E9:7_1_2"><i><font color="Green">E9</font></i></a>, <a class="txt" href="bhsp_5.html#E16:7_1_2"><i><font color="Green">E14</font></i></a>, <a class="ref" href="bhsp_5.html#T1" target="_self">Th1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><a NAME="E1:7_1_2_1"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>10</sub></font> implies <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="bhsp_5.html#E24:7_1_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><b>hence </b><a NAME="E2:7_1_2_1"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> iff <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="bhsp_5.html#E25:7_1_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/></div><b>end;</b></div>
<a NAME="E28:7_1_2"/><b>then </b><i><font color="Green">E22</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<a NAME="E29:7_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>10</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:7_1_2_2"/><i><font color="Green">E23</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>9</sub></font> is   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_2_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:7_1_2_2_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="bhsp_5.html#E1:7_1_2_2"><i><font color="Green">E23</font></i></a></i>;<br/><a NAME="E3:7_1_2_2_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="bhsp_5.html#E25:7_1_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="bhsp_5.html#E1:7_1_2_2_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="bhsp_5.html#E1:7_1_2_2"><i><font color="Green">E23</font></i></a></i>;<br/><a NAME="E5:7_1_2_2_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="bhsp_5.html#E12:7_1_2"><i><font color="Green">E11</font></i></a>, <a class="txt" href="bhsp_5.html#E25:7_1_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="bhsp_5.html#E26:7_1_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="bhsp_5.html#E1:7_1_2_2_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><a NAME="E6:7_1_2_2_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="bhsp_5.html#E16:7_1_2"><i><font color="Green">E14</font></i></a>, <a class="txt" href="bhsp_5.html#E19:7_1_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="bhsp_5.html#E1:7_1_2_2_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="funct_2.html#T67">FUNCT_2:67</a></i>;<br/><font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> = 
 <a href="bhsp_5.html#K3" target="_self">PO</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="bhsp_5.html#E17:7_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="bhsp_5.html#E1:7_1_2_2_1"><i><font color="Green">E24</font></i></a></i>
<br/>.= 
the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="bhsp_5.html#E21:7_1_2"><i><font color="Green">E18</font></i></a></i>
<br/>.= 
the <a href="bhsp_1.html#U1">scalar</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="txt" href="bhsp_5.html#E6:7_1_2_2_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
 <a href="bhsp_5.html#K3" target="_self">PO</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>2</sub></font>

<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="bhsp_5.html#E13:7_1_2"><i><font color="Green">E12</font></i></a>, <a class="txt" href="bhsp_5.html#E5:7_1_2_2_1"><i><font color="Green">E25</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="txt" href="bhsp_5.html#E6:7_1_2_2_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/><b>hence </b><a NAME="E8:7_1_2_2_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:7_1_2_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="bhsp_5.html#E28:7_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E30:7_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="bhsp_5.html#E12:7_1_2"><i><font color="Green">E11</font></i></a>, <a class="txt" href="bhsp_5.html#E14:7_1_2"><i><font color="Green">E13</font></i></a>, <a class="txt" href="bhsp_5.html#E18:7_1_2"><i><font color="Green">E16</font></i></a>, <a class="ref" href="finsop_1.html#T8">FINSOP_1:8</a></i>;<br/>


</div>
