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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="orders_3.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font>,   <a href="orders_3.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font>,   <a href="hidden.html#M1">set</a> ] means <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="orders_3.html#K1" target="_self">MonFuncs</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>2</sub></font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="funct_1.html#NM1">Function</a>,  <a href="funct_1.html#NM1">Function</a>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">a<sub>1</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">a<sub>2</sub></font>;<br/>
<a NAME="E1:35_1_1"/><i><font color="Green">E19</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="orders_3.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="ens_1.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="orders_3.html#K2" target="_self">Carr</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>5</sub></font>] holds <br/>( <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>5</sub></font>,<font color="Olive">b<sub>4</sub></font>) <a href="hidden.html#R2">in</a>  <a href="ens_1.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="orders_3.html#K2" target="_self">Carr</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>5</sub></font>,<font color="Olive">b<sub>4</sub></font>)] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <a href="orders_3.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="ens_1.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="orders_3.html#K2" target="_self">Carr</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:35_1_1_1"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="orders_3.html#K1" target="_self">MonFuncs</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="orders_3.html#K1" target="_self">MonFuncs</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 ;<br/>

<a NAME="E2:35_1_1_1"/><i><font color="Green">E21</font></i>: 
 <a href="orders_3.html#K1" target="_self">MonFuncs</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="ens_1.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="orders_3.html#K2" target="_self">Carr</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="orders_3.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E3:35_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="orders_3.html#K1" target="_self">MonFuncs</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="orders_3.html#E1:35_1_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="orders_3.html#T6" target="_self">Th6</a></i>;<br/>
<b>hence </b><a NAME="E4:35_1_1_1"/>
( <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font>) <a href="hidden.html#R2">in</a>  <a href="ens_1.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="orders_3.html#K2" target="_self">Carr</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font>)] )
 <b>by </b><i><a class="txt" href="orders_3.html#E2:35_1_1_1"><i><font color="Green">E21</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:35_1_1"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="orders_3.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="ens_1.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="orders_3.html#K2" target="_self">Carr</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] &amp; ( for <font color="Olive">b<sub>3</sub></font> being   <a href="orders_3.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>4</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="ens_1.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="orders_3.html#K2" target="_self">Carr</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> holds <br/>( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font>] implies <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>2</sub></font>) <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font> ) &amp; ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>4</sub></font>] implies <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font>) <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font> ) ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="orders_3.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E1:35_1_1_2"/><i><font color="Green">E21</font></i>: 
 <a href="orders_3.html#K1" target="_self">MonFuncs</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="ens_1.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="orders_3.html#K2" target="_self">Carr</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="orders_3.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E2:35_1_1_2"/>
 <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="orders_3.html#K1" target="_self">MonFuncs</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="orders_3.html#T7" target="_self">Th7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> =  <a href="partfun1.html#K6">id</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="ens_1.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="orders_3.html#K2" target="_self">Carr</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="orders_3.html#E1:35_1_1_2"><i><font color="Green">E21</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="orders_3.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="ens_1.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="orders_3.html#K2" target="_self">Carr</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/><b>thus </b><a NAME="E1:35_1_1_2_1"/>
( ( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="orders_3.html#K1" target="_self">MonFuncs</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> implies <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> ) &amp; ( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="orders_3.html#K1" target="_self">MonFuncs</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font> implies <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <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="35_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:35_1_1_2_1_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="orders_3.html#K1" target="_self">MonFuncs</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> implies <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">c<sub>4</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="35_1_1_2_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:35_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="orders_3.html#K1" target="_self">MonFuncs</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font><b> such that </b><br/><a NAME="E3:35_1_1_2_1_1_1"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> is <a href="orders_3.html#V5" target="_self">monotone</a> )
 <b>by </b><i><a class="ref" href="orders_3.html#D6" target="_self">Def6</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font> <b>by </b><i><a class="txt" href="orders_3.html#E3:35_1_1_2_1_1_1"><i><font color="Green">E22</font></i></a></i>;<br/>
<a NAME="E5:35_1_1_2_1_1_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:35_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <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="ref" href="relat_1.html#T77">RELAT_1:77</a></i>;<br/>


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

<b>thus </b><a NAME="E2:35_1_1_2_1_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="orders_3.html#K1" target="_self">MonFuncs</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font> implies <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <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="35_1_1_2_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:35_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="orders_3.html#K1" target="_self">MonFuncs</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:35_1_1_2_1_1_2"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> is <a href="orders_3.html#V5" target="_self">monotone</a> )
 <b>by </b><i><a class="ref" href="orders_3.html#D6" target="_self">Def6</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="orders_3.html#E3:35_1_1_2_1_1_2"><i><font color="Green">E22</font></i></a></i>;<br/>
<a NAME="E5:35_1_1_2_1_1_2"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>
 
;<br/>
<b>hence </b><a NAME="E6:35_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <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="ref" href="relat_1.html#T79">RELAT_1:79</a></i>;<br/>


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


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:35_1_1_2"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>] &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="orders_3.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="ens_1.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="orders_3.html#K2" target="_self">Carr</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> holds <br/>( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>] implies <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>) <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> ) &amp; ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>2</sub></font>] implies <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>4</sub></font>,<font color="Olive">b<sub>2</sub></font>) <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> ) ) ) )
 <b>by </b><i><a class="ref" href="orders_3.html#T7" target="_self">Th7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:35_1_1"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being   <a href="orders_3.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>5</sub></font>, <font color="Olive">b<sub>6</sub></font>, <font color="Olive">b<sub>7</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="ens_1.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="orders_3.html#K2" target="_self">Carr</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>5</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>6</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>7</sub></font>] holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>7</sub></font>,<font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>6</sub></font>,<font color="Olive">b<sub>5</sub></font>)) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>7</sub></font>,<font color="Olive">b<sub>6</sub></font>),<font color="Olive">b<sub>5</sub></font>)
 
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>;<br/>
<a NAME="E4:35_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being <a href="cat_1.html#V1">strict</a> <a href="cat_5.html#V1">with_triple-like_morphisms</a> <a href="cat_1.html#NM4">Category</a> st <br/>( the <a href="cat_1.html#U1">Objects</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="orders_3.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>4</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="ens_1.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="orders_3.html#K2" target="_self">Carr</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font>] holds <br/><span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Olive">b<sub>4</sub></font></span><a href="domain_1.html#K1">]</a></span> is   <a href="cat_1.html#NM2">Morphism</a> of <font color="Olive">b<sub>1</sub></font> ) &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Olive">b<sub>1</sub></font>ex <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being   <a href="orders_3.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>5</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="ens_1.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="orders_3.html#K2" target="_self">Carr</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Olive">b<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font>] ) ) &amp; ( for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Olive">b<sub>1</sub></font><br/>for <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font>, <font color="Olive">b<sub>6</sub></font> being   <a href="orders_3.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>7</sub></font>, <font color="Olive">b<sub>8</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="ens_1.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="orders_3.html#K2" target="_self">Carr</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Olive">b<sub>7</sub></font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>5</sub></font>,<font color="Olive">b<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Olive">b<sub>8</sub></font></span><a href="domain_1.html#K1">]</a></span> holds <br/><font color="Olive">b<sub>3</sub></font> <a href="cat_1.html#K4">*</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>8</sub></font>,<font color="Olive">b<sub>7</sub></font>)</span><a href="tarski.html#K4">]</a></span> ) )
 
<b>from </b><i><a class="ref" href="cat_5.html#S1">CAT_5:sch 1</a>(<a class="txt" href="orders_3.html#E1:35_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="orders_3.html#E2:35_1_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="orders_3.html#E3:35_1_1"><i><font color="Green">E21</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:35_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being <a href="cat_1.html#V1">strict</a> <a href="cat_5.html#V1">with_triple-like_morphisms</a> <a href="cat_1.html#NM4">Category</a> st <br/>( the <a href="cat_1.html#U1">Objects</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="orders_3.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>4</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="ens_1.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="orders_3.html#K2" target="_self">Carr</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="orders_3.html#K1" target="_self">MonFuncs</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> holds <br/><span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Olive">b<sub>4</sub></font></span><a href="domain_1.html#K1">]</a></span> is   <a href="cat_1.html#NM2">Morphism</a> of <font color="Olive">b<sub>1</sub></font> ) &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Olive">b<sub>1</sub></font>ex <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being   <a href="orders_3.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>5</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="ens_1.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="orders_3.html#K2" target="_self">Carr</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Olive">b<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="orders_3.html#K1" target="_self">MonFuncs</a> <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font> ) ) &amp; ( for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="cat_1.html#NM2">Morphism</a> of <font color="Olive">b<sub>1</sub></font><br/>for <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font>, <font color="Olive">b<sub>6</sub></font> being   <a href="orders_3.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>7</sub></font>, <font color="Olive">b<sub>8</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="ens_1.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="orders_3.html#K2" target="_self">Carr</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Olive">b<sub>7</sub></font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>5</sub></font>,<font color="Olive">b<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span>,<font color="Olive">b<sub>8</sub></font></span><a href="domain_1.html#K1">]</a></span> holds <br/><font color="Olive">b<sub>3</sub></font> <a href="cat_1.html#K4">*</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span>,<span class="p2">(<span class="default"><font color="Olive">b<sub>8</sub></font> <a href="funct_1.html#NK3" target="_self">*</a> <font color="Olive">b<sub>7</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> ) )
 ;<br/>


</div>
