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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="binop_1.html#NM2">BinOp</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K1">REAL</a> ;<br/>



<b>assume </b><a NAME="E1:47"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="pscomp_1.html#NM1">RealMap</a> of <font color="Maroon">c<sub>1</sub></font> holds <font color="Maroon">c<sub>2</sub></font> <a href="binop_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="nagata_1.html#K3" target="_self">+</a> <font color="Olive">b<sub>2</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K1">REAL</a> ;<br/>
<b>thus </b><a NAME="E2:47"/>
<font color="Maroon">c<sub>2</sub></font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>proof </b></a><div class="add">

<a NAME="E1:47_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K1">REAL</a>  st <font color="Olive">b<sub>1</sub></font> <a href="binop_1.html#R3">is_a_unity_wrt</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="borsuk_1.html#K1">--&gt;</a> 0 as   <a href="pscomp_1.html#NM1">RealMap</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="funcop_1.html#T57">FUNCOP_1:57</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K1">REAL</a> ;<br/><a NAME="E1:47_1_1_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E1:47_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nagata_1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="nagata_1.html#D7" target="_self">Def7</a></i>;<br/><a NAME="E2:47_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nagata_1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 0 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><b>hence </b><a NAME="E3:47_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nagata_1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <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_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/></div><b>end;</b></div>
<a NAME="E2:47_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="nagata_1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>
<b>hence </b><a NAME="E3:47_1_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="nagata_1.html#E1:47"><i><font color="Green">E25</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E2:47_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_1_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E1:47_1_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nagata_1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="nagata_1.html#D7" target="_self">Def7</a></i>;<br/><a NAME="E2:47_1_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nagata_1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 0
 <b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><b>hence </b><a NAME="E3:47_1_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nagata_1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <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_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/></div><b>end;</b></div>
<a NAME="E2:47_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="nagata_1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>
<b>hence </b><a NAME="E3:47_1_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="nagata_1.html#E1:47"><i><font color="Green">E25</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:47_1_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="nagata_1.html#E1:47_1_1_1"><i><font color="Green">E26</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:47_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="binop_1.html#R3">is_a_unity_wrt</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="binop_1.html#T11">BINOP_1:11</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:47_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="setwiseo.html#NR1" target="_self">has_a_unity</a> 
 <b>by </b><i><a class="ref" href="setwiseo.html#D2">SETWISEO:def 2</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:47"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="binop_1.html#V1">commutative</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2"><b>proof </b></a><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="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K1">REAL</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D13">BINOP_1:def 13</a><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E1:47_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nagata_1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="nagata_1.html#D7" target="_self">Def7</a></i>;<br/><b>hence </b><a NAME="E2:47_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nagata_1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nagata_1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="nagata_1.html#D7" target="_self">Def7</a></i>;<br/></div><b>end;</b></div>
<a NAME="E2:47_2"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="nagata_1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="nagata_1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="nagata_1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="nagata_1.html#E1:47"><i><font color="Green">E25</font></i></a>, <a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>
<b>hence </b><a NAME="E3:47_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="nagata_1.html#E1:47"><i><font color="Green">E25</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E4:47"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="binop_1.html#V2">associative</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="numbers.html#K1">REAL</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="binop_1.html#D14">BINOP_1:def 14</a><br/>

<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font> as   <a href="pscomp_1.html#NM1">RealMap</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="funct_2.html#T121">FUNCT_2:121</a></i>;<br/>
<a NAME="E2:47_3"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="nagata_1.html#K3" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nagata_1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nagata_1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nagata_1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>6</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_3_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_3_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><a NAME="E1:47_3_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nagata_1.html#K3" target="_self">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nagata_1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nagata_1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nagata_1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="nagata_1.html#D7" target="_self">Def7</a></i>;<br/><a NAME="E2:47_3_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nagata_1.html#K3" target="_self">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nagata_1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 ;<br/><a NAME="E3:47_3_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nagata_1.html#K3" target="_self">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nagata_1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nagata_1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="nagata_1.html#D7" target="_self">Def7</a></i>;<br/><b>hence </b><a NAME="E4:47_3_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nagata_1.html#K3" target="_self">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nagata_1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nagata_1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nagata_1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="nagata_1.html#D7" target="_self">Def7</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:47_3_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="nagata_1.html#K3" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nagata_1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nagata_1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nagata_1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:47_3"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="nagata_1.html#K3" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nagata_1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="nagata_1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>8</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nagata_1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="nagata_1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="nagata_1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="nagata_1.html#E1:47"><i><font color="Green">E25</font></i></a></i>;<br/>
<a NAME="E4:47_3"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="nagata_1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="nagata_1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="nagata_1.html#K3" target="_self">+</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="nagata_1.html#E1:47"><i><font color="Green">E25</font></i></a>, <a class="txt" href="nagata_1.html#E2:47_3"><i><font color="Green">E26</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:47_3"/>
<font color="Maroon">c<sub>2</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="nagata_1.html#E1:47"><i><font color="Green">E25</font></i></a></i>;<br/>


</div><b>end;</b></div>


</div>
