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

<b>consider </b><font color="Maroon">c<sub>1</sub></font> being    <a href="hidden.html#M1">set</a> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>2</sub></font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="lang1.html#K2">&lt;*&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">c<sub>3</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="partfun1.html#K4">PFuncs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_2.html#K3">*</a> </span>)</span>,<font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="ref" href="unialg_1.html#T3">UNIALG_1:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="lang1.html#K3">*&gt;</a></span> as   <a href="unialg_1.html#NM1">PFuncFinSequence</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="unialg_1.html#G1">UAStr</a>(# <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> #);<br/>
<a NAME="E6:11_1_1"/><i><font color="Green">E8</font></i>: 
( the <a href="unialg_1.html#U1">charact</a> of <a href="unialg_1.html#G1">UAStr</a>(# <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> #) is <a href="unialg_1.html#V5">quasi_total</a> &amp; the <a href="unialg_1.html#U1">charact</a> of <a href="unialg_1.html#G1">UAStr</a>(# <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> #) is <a href="unialg_1.html#V4">homogeneous</a> &amp; not the <a href="unialg_1.html#U1">charact</a> of <a href="unialg_1.html#G1">UAStr</a>(# <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> #) is <a href="relat_1.html#V2">with_zero</a> )
 
<b>by </b><i><a class="ref" href="unialg_1.html#T4">UNIALG_1:4</a></i>;<br/>
<a NAME="E7:11_1_1"/><i><font color="Green">E9</font></i>: 
 <a href="finseq_1.html#K3">len</a> the <a href="unialg_1.html#U1">charact</a> of <a href="unialg_1.html#G1">UAStr</a>(# <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> #) <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> =  <a href="unialg_1.html#G1">UAStr</a>(# <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> #) as   <a href="unialg_1.html#NM2">Universal_Algebra</a> <b>by </b><i><a class="txt" href="freealg.html#E6:11_1_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="unialg_1.html#D7">UNIALG_1:def 7</a>, <a class="ref" href="unialg_1.html#D8">UNIALG_1:def 8</a>, <a class="ref" href="unialg_1.html#D9">UNIALG_1:def 9</a></i>;<br/>
<a NAME="E9:11_1_1"/><i><font color="Green">E10</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span> &amp; 1 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="txt" href="freealg.html#E7:11_1_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> 1 as   <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">c<sub>7</sub></font> <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E11:11_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>
<a NAME="E12:11_1_1"/><b>then </b><i><font color="Green">E11</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="lang1.html#K2">&lt;*&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>assume </b><a NAME="E1:11_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><a NAME="E2:11_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="lang1.html#K2">&lt;*&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="freealg.html#E12:11_1_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:11_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="finseq_1.html#T32">FINSEQ_1:32</a></i>;<br/></div><b>end;</b></div>
<a NAME="E14:11_1_1"/><b>then </b><i><font color="Green">E12</font></i>: 
 <a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="unialg_1.html#D10">UNIALG_1:def 10</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>7</sub></font>
;<br/>

<a NAME="E15:11_1_1"/>
 <a href="unialg_2.html#K7">GenUnivAlg</a> <span class="p1">(<span class="default"><a href="subset_1.html#K1">{}</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_2"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">c<sub>9</sub></font> =  <a href="subset_1.html#K1">{}</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K7">GenUnivAlg</a> <span class="p2">(<span class="default"><a href="subset_1.html#K1">{}</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>7</sub></font> <b>by </b><i><a class="ref" href="unialg_2.html#D8">UNIALG_2:def 8</a></i>;<br/>
<a NAME="E2:11_1_1_2"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T39">ZFMISC_1:39</a></i>;<br/>
<a NAME="E3:11_1_1_2"/><i><font color="Green">E14</font></i>: 
 <a href="finseq_1.html#K4">dom</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="unialg_2.html#K3">Opers</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D7">UNIALG_2:def 7</a></i>;<br/>
<a NAME="E4:11_1_1_2"/>
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="finseq_1.html#K4">dom</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>7</sub></font> holds <br/>the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="unialg_2.html#K3">Opers</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:11_1_1_2_1"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>7</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font> as   <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">c<sub>7</sub></font> <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E3:11_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="unialg_2.html#K3">Opers</a> <font color="Maroon">c<sub>7</sub></font>,<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> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="unialg_2.html#K2">/.</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="freealg.html#E3:11_1_1_2"><i><font color="Green">E14</font></i></a>, <a class="txt" href="freealg.html#E1:11_1_1_2_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="unialg_2.html#D7">UNIALG_2:def 7</a></i>;<br/>
<b>hence </b><a NAME="E4:11_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="unialg_2.html#K3">Opers</a> <font color="Maroon">c<sub>7</sub></font>,<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> <a href="hidden.html#R1">=</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="freealg.html#E2:11_1_1_2"><i><font color="Green">E13</font></i></a>, <a class="ref" href="unialg_2.html#T7">UNIALG_2:7</a></i>;<br/>


</div><b>end;</b></div>
<b>then </b>the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>7</sub></font> = 
 <a href="unialg_2.html#K3">Opers</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="freealg.html#E3:11_1_1_2"><i><font color="Green">E14</font></i></a>, <a class="ref" href="finseq_1.html#T17">FINSEQ_1:17</a></i>
<br/>.= 
the <a href="unialg_1.html#U1">charact</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K7">GenUnivAlg</a> <span class="p2">(<span class="default"><a href="subset_1.html#K1">{}</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="unialg_2.html#D8">UNIALG_2:def 8</a></i>
;<br/>
<b>hence </b><a NAME="E6:11_1_1_2"/>
 <a href="unialg_2.html#K7">GenUnivAlg</a> <span class="p1">(<span class="default"><a href="subset_1.html#K1">{}</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="freealg.html#E2:11_1_1_2"><i><font color="Green">E13</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> =  <a href="subset_1.html#K1">{}</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font> as    <a href="freealg.html#M1" target="_self">GeneratorSet</a> of <font color="Maroon">c<sub>7</sub></font> <b>by </b><i><a class="ref" href="freealg.html#D5" target="_self">Def5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_3"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>9</sub></font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>11</sub></font> be   <a href="unialg_1.html#NM2">Universal_Algebra</a>;<br/><b>assume </b><a NAME="E1:11_1_1_3_1"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="unialg_2.html#R1">are_similar</a> 
 ;<br/><a NAME="E2:11_1_1_3_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="freealg.html#E7:11_1_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="unialg_2.html#T3">UNIALG_2:3</a></i>;<br/><a NAME="E3:11_1_1_3_1"/><b>then </b><i><font color="Green">E14</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span> &amp; 1 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> 1 as   <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">c<sub>11</sub></font> <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:11_1_1_3_1"/><i><font color="Green">E15</font></i>: 
 <a href="unialg_1.html#K3">signature</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K3">signature</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="freealg.html#E1:11_1_1_3_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="unialg_2.html#D2">UNIALG_2:def 2</a></i>;<br/><a NAME="E6:11_1_1_3_1"/>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K3">signature</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>7</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K3">signature</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>11</sub></font> &amp;  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K3">signature</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="unialg_1.html#K3">signature</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K3">signature</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="unialg_1.html#K3">signature</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a>, <a class="ref" href="unialg_1.html#D11">UNIALG_1:def 11</a></i>;<br/><a NAME="E7:11_1_1_3_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="unialg_1.html#K3">signature</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>8</sub></font> &amp; <span class="p1">(<span class="default"><a href="unialg_1.html#K3">signature</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="freealg.html#E7:11_1_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="freealg.html#E3:11_1_1_3_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="freealg.html#E5:11_1_1_3_1"><i><font color="Green">E15</font></i></a>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="unialg_1.html#D11">UNIALG_1:def 11</a></i>;<br/><a NAME="E8:11_1_1_3_1"/><b>then </b><i><font color="Green">E16</font></i>: 
 <a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="unialg_1.html#K2">arity</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="freealg.html#E1:11_1_1_3_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="unialg_2.html#D2">UNIALG_2:def 2</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E10:11_1_1_3_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><a NAME="E11:11_1_1_3_1"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>12</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="txt" href="freealg.html#E10:11_1_1_3_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>13</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>11</sub></font> ;<br/><b>let </b><font color="Maroon">c<sub>15</sub></font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>10</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>11</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">c<sub>14</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>11</sub></font> <b>by </b><i><a class="ref" href="funcop_1.html#T57">FUNCOP_1:57</a></i>;<br/><a NAME="E14:11_1_1_3_1"/><i><font color="Green">E18</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>7</sub></font> holds <font color="Maroon">c<sub>16</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/><b>take </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>16</sub></font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="11_1_1_3_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>18</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:11_1_1_3_1_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>7</sub></font>
 ;<br/><a NAME="E2:11_1_1_3_1_1"/><b>then </b><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="freealg.html#E9:11_1_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>let </b><font color="Maroon">c<sub>19</sub></font> be   <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">c<sub>7</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>20</sub></font> be   <a href="unialg_2.html#NM5">operation</a> of <font color="Maroon">c<sub>11</sub></font>;<br/><b>assume </b><a NAME="E3:11_1_1_3_1_1"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font> )
 ;<br/><b>let </b><font color="Maroon">c<sub>21</sub></font> be   <a href="struct_0.html#NM8">FinSequence</a> of <font color="Maroon">c<sub>7</sub></font>;<br/><b>assume </b><a NAME="E4:11_1_1_3_1_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>19</sub></font>
 ;<br/><a NAME="E5:11_1_1_3_1_1"/><b>then </b>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a>  <a href="lang1.html#K2">&lt;*&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="freealg.html#E12:11_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="freealg.html#E2:11_1_1_3_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="freealg.html#E3:11_1_1_3_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E6:11_1_1_3_1_1"/><b>then </b><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="lang1.html#K15">*</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a>  <a href="lang1.html#K2">&lt;*&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="ref" href="alg_1.html#T3">ALG_1:3</a></i>;<br/> <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>20</sub></font> = 
0 <a href="finseq_2.html#K4">-tuples_on</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="txt" href="freealg.html#E14:11_1_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="freealg.html#E8:11_1_1_3_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="freealg.html#E2:11_1_1_3_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="freealg.html#E3:11_1_1_3_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="unialg_2.html#T2">UNIALG_2:2</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="lang1.html#K2">&lt;*&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="ref" href="finseq_2.html#T112">FINSEQ_2:112</a></i>
;<br/><a NAME="E8:11_1_1_3_1_1"/><b>then </b><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="lang1.html#K15">*</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="freealg.html#E10:11_1_1_3_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="freealg.html#E2:11_1_1_3_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="freealg.html#E3:11_1_1_3_1_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="freealg.html#E6:11_1_1_3_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E9:11_1_1_3_1_1"/>
( <font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>19</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>19</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="txt" href="freealg.html#E4:11_1_1_3_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/><b>hence </b><a NAME="E10:11_1_1_3_1_1"/>
<font color="Maroon">c<sub>17</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="lang1.html#K15">*</a> <font color="Maroon">c<sub>21</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="freealg.html#E14:11_1_1_3_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="freealg.html#E8:11_1_1_3_1_1"><i><font color="Green">E23</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E16:11_1_1_3_1"/>
<font color="Maroon">c<sub>17</sub></font> <a href="alg_1.html#R1">is_homomorphism</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="freealg.html#E1:11_1_1_3_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="alg_1.html#D1">ALG_1:def 1</a></i>;<br/><a NAME="E17:11_1_1_3_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="partfun1.html#T54">PARTFUN1:54</a></i>;<br/><a NAME="E18:11_1_1_3_1"/><b>then </b>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="relat_1.html#T64">RELAT_1:64</a></i>;<br/><b>hence </b><a NAME="E19:11_1_1_3_1"/>
<font color="Maroon">c<sub>17</sub></font> <a href="partfun1.html#K2">|</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="ref" href="relat_1.html#T110">RELAT_1:110</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:11_1_1_3"/>
<font color="Maroon">c<sub>10</sub></font> is <a href="freealg.html#V2" target="_self">free</a>
 <b>by </b><i><a class="ref" href="freealg.html#D6" target="_self">Def6</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E18:11_1_1"/>
( <font color="Maroon">c<sub>7</sub></font> is <a href="freealg.html#V3" target="_self">free</a> &amp; <font color="Maroon">c<sub>7</sub></font> is <a href="unialg_1.html#V3">strict</a> )
 <b>by </b><i><a class="ref" href="freealg.html#D7" target="_self">Def7</a></i>;<br/>


</div>
