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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="unialg_1.html#NM2">Universal_Algebra</a>;<br/>



<b>assume </b><a NAME="E1:5"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="unialg_2.html#R1">are_similar</a> 
 ;<br/>

<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="unialg_1.html#G1">UAStr</a>(# <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K4" target="_self">Opers</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> #);<br/>
<a NAME="E2:5"/><i><font color="Green">E8</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K4" target="_self">Opers</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</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="pralg_1.html#K4" target="_self">Opers</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>1</sub></font> <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> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="finseq_1.html#K4">dom</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>2</sub></font> <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> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E3:5"/><i><font color="Green">E9</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K4" target="_self">Opers</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</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>1</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>2</sub></font> <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>1</sub></font> )
 
<b>by </b><i><a class="txt" href="pralg_1.html#E1:5"><i><font color="Green">E7</font></i></a>, <a class="ref" href="pralg_1.html#D4" target="_self">Def4</a>, <a class="ref" href="unialg_2.html#T3">UNIALG_2:3</a></i>;<br/>
<a NAME="E4:5"/><i><font color="Green">E10</font></i>: 
the <a href="unialg_1.html#U1">charact</a> of <a href="unialg_1.html#G1">UAStr</a>(# <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K4" target="_self">Opers</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> #) is <a href="unialg_1.html#V5">quasi_total</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="unialg_1.html#D5">UNIALG_1:def 5</a><br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="partfun1.html#NM1">PartFunc</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="unialg_1.html#G1">UAStr</a>(# <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K4" target="_self">Opers</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> #) <a href="finseq_2.html#K3">*</a> ,the <a href="struct_0.html#U1">carrier</a> of <a href="unialg_1.html#G1">UAStr</a>(# <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K4" target="_self">Opers</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> #);<br/>



<b>assume </b><a NAME="E1:5_1"/><i><font color="Green">E11</font></i>: 
( <font color="Maroon">c<sub>4</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 <a href="unialg_1.html#G1">UAStr</a>(# <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K4" target="_self">Opers</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> #) &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> the <a href="unialg_1.html#U1">charact</a> of <a href="unialg_1.html#G1">UAStr</a>(# <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K4" target="_self">Opers</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> #) <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="unialg_1.html#V1">homogeneous</a> <a href="unialg_1.html#V2">quasi_total</a> <a href="partfun1.html#NM1">PartFunc</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K3">*</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="pralg_1.html#E2:5"><i><font color="Green">E8</font></i></a>, <a class="txt" href="pralg_1.html#E3:5"><i><font color="Green">E9</font></i></a>, <a class="ref" href="funct_1.html#D15">FUNCT_1:def 15</a>, <a class="ref" href="unialg_1.html#T5">UNIALG_1:5</a>, <a class="ref" href="unialg_1.html#D4">UNIALG_1:def 4</a>, <a class="ref" href="unialg_1.html#D5">UNIALG_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="unialg_1.html#V1">homogeneous</a> <a href="unialg_1.html#V2">quasi_total</a> <a href="partfun1.html#NM1">PartFunc</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="finseq_2.html#K3">*</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="txt" href="pralg_1.html#E2:5"><i><font color="Green">E8</font></i></a>, <a class="txt" href="pralg_1.html#E3:5"><i><font color="Green">E9</font></i></a>, <a class="txt" href="pralg_1.html#E1:5_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="funct_1.html#D15">FUNCT_1:def 15</a>, <a class="ref" href="unialg_1.html#T5">UNIALG_1:5</a>, <a class="ref" href="unialg_1.html#D4">UNIALG_1:def 4</a>, <a class="ref" href="unialg_1.html#D5">UNIALG_1:def 5</a></i>;<br/>
<a NAME="E4:5_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="pralg_1.html#K3" target="_self">[[:</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="pralg_1.html#K3" target="_self">:]]</a></span>
 
<b>by </b><i><a class="txt" href="pralg_1.html#E1:5"><i><font color="Green">E7</font></i></a>, <a class="txt" href="pralg_1.html#E1:5_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="pralg_1.html#D4" target="_self">Def4</a></i>;<br/>
<b>hence </b><a NAME="E5:5_1"/>
<font color="Maroon">c<sub>5</sub></font> is <a href="unialg_1.html#V2">quasi_total</a>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:5"/><i><font color="Green">E11</font></i>: 
the <a href="unialg_1.html#U1">charact</a> of <a href="unialg_1.html#G1">UAStr</a>(# <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K4" target="_self">Opers</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> #) is <a href="unialg_1.html#V4">homogeneous</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="unialg_1.html#D4">UNIALG_1:def 4</a><br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="partfun1.html#NM1">PartFunc</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="unialg_1.html#G1">UAStr</a>(# <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K4" target="_self">Opers</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> #) <a href="finseq_2.html#K3">*</a> ,the <a href="struct_0.html#U1">carrier</a> of <a href="unialg_1.html#G1">UAStr</a>(# <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K4" target="_self">Opers</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> #);<br/>



<b>assume </b><a NAME="E1:5_2"/><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>4</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 <a href="unialg_1.html#G1">UAStr</a>(# <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K4" target="_self">Opers</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> #) &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> the <a href="unialg_1.html#U1">charact</a> of <a href="unialg_1.html#G1">UAStr</a>(# <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K4" target="_self">Opers</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> #) <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="unialg_1.html#V1">homogeneous</a> <a href="unialg_1.html#V2">quasi_total</a> <a href="partfun1.html#NM1">PartFunc</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K3">*</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="pralg_1.html#E2:5"><i><font color="Green">E8</font></i></a>, <a class="txt" href="pralg_1.html#E3:5"><i><font color="Green">E9</font></i></a>, <a class="ref" href="funct_1.html#D15">FUNCT_1:def 15</a>, <a class="ref" href="unialg_1.html#T5">UNIALG_1:5</a>, <a class="ref" href="unialg_1.html#D4">UNIALG_1:def 4</a>, <a class="ref" href="unialg_1.html#D5">UNIALG_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="unialg_1.html#V1">homogeneous</a> <a href="unialg_1.html#V2">quasi_total</a> <a href="partfun1.html#NM1">PartFunc</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="finseq_2.html#K3">*</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="txt" href="pralg_1.html#E2:5"><i><font color="Green">E8</font></i></a>, <a class="txt" href="pralg_1.html#E3:5"><i><font color="Green">E9</font></i></a>, <a class="txt" href="pralg_1.html#E1:5_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="funct_1.html#D15">FUNCT_1:def 15</a>, <a class="ref" href="unialg_1.html#T5">UNIALG_1:5</a>, <a class="ref" href="unialg_1.html#D4">UNIALG_1:def 4</a>, <a class="ref" href="unialg_1.html#D5">UNIALG_1:def 5</a></i>;<br/>
<a NAME="E4:5_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="pralg_1.html#K3" target="_self">[[:</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="pralg_1.html#K3" target="_self">:]]</a></span>
 
<b>by </b><i><a class="txt" href="pralg_1.html#E1:5"><i><font color="Green">E7</font></i></a>, <a class="txt" href="pralg_1.html#E1:5_2"><i><font color="Green">E12</font></i></a>, <a class="ref" href="pralg_1.html#D4" target="_self">Def4</a></i>;<br/>
<b>hence </b><a NAME="E5:5_2"/>
<font color="Maroon">c<sub>5</sub></font> is <a href="unialg_1.html#V1">homogeneous</a>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:5"/><i><font color="Green">E12</font></i>: 
the <a href="unialg_1.html#U1">charact</a> of <a href="unialg_1.html#G1">UAStr</a>(# <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K4" target="_self">Opers</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> #) <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:5_3"/>
the <a href="unialg_1.html#U1">charact</a> of <a href="unialg_1.html#G1">UAStr</a>(# <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K4" target="_self">Opers</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> #) <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<a NAME="E2:5_3"/><b>then </b>
 <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>(# <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K4" target="_self">Opers</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> #) <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<b>hence </b><a NAME="E3:5_3"/>
contradiction
 <b>by </b><i><a class="txt" href="pralg_1.html#E3:5"><i><font color="Green">E9</font></i></a>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:5"/>
the <a href="unialg_1.html#U1">charact</a> of <a href="unialg_1.html#G1">UAStr</a>(# <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K4" target="_self">Opers</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> #) is <a href="relat_1.html#V2">non-empty</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D15">FUNCT_1:def 15</a><br/>

<b>set </b><font color="Maroon">c<sub>5</sub></font> = the <a href="unialg_1.html#U1">charact</a> of <a href="unialg_1.html#G1">UAStr</a>(# <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K4" target="_self">Opers</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> #) <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<b>assume </b><a NAME="E1:5_4"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>4</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 <a href="unialg_1.html#G1">UAStr</a>(# <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K4" target="_self">Opers</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> #)
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = the <a href="unialg_1.html#U1">charact</a> of <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="unialg_1.html#V1">homogeneous</a> <a href="unialg_1.html#V2">quasi_total</a> <a href="partfun1.html#NM1">PartFunc</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <a href="finseq_2.html#K3">*</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="pralg_1.html#E2:5"><i><font color="Green">E8</font></i></a>, <a class="txt" href="pralg_1.html#E3:5"><i><font color="Green">E9</font></i></a>, <a class="txt" href="pralg_1.html#E1:5_4"><i><font color="Green">E13</font></i></a>, <a class="ref" href="funct_1.html#D15">FUNCT_1:def 15</a>, <a class="ref" href="unialg_1.html#T5">UNIALG_1:5</a>, <a class="ref" href="unialg_1.html#D4">UNIALG_1:def 4</a>, <a class="ref" href="unialg_1.html#D5">UNIALG_1:def 5</a></i>;<br/>
<b>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>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="unialg_1.html#V1">homogeneous</a> <a href="unialg_1.html#V2">quasi_total</a> <a href="partfun1.html#NM1">PartFunc</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font> <a href="finseq_2.html#K3">*</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="txt" href="pralg_1.html#E2:5"><i><font color="Green">E8</font></i></a>, <a class="txt" href="pralg_1.html#E3:5"><i><font color="Green">E9</font></i></a>, <a class="txt" href="pralg_1.html#E1:5_4"><i><font color="Green">E13</font></i></a>, <a class="ref" href="funct_1.html#D15">FUNCT_1:def 15</a>, <a class="ref" href="unialg_1.html#T5">UNIALG_1:5</a>, <a class="ref" href="unialg_1.html#D4">UNIALG_1:def 4</a>, <a class="ref" href="unialg_1.html#D5">UNIALG_1:def 5</a></i>;<br/>
<a NAME="E5:5_4"/>
the <a href="unialg_1.html#U1">charact</a> of <a href="unialg_1.html#G1">UAStr</a>(# <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K4" target="_self">Opers</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> #) <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="pralg_1.html#K3" target="_self">[[:</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="pralg_1.html#K3" target="_self">:]]</a></span>
 
<b>by </b><i><a class="txt" href="pralg_1.html#E1:5"><i><font color="Green">E7</font></i></a>, <a class="txt" href="pralg_1.html#E1:5_4"><i><font color="Green">E13</font></i></a>, <a class="ref" href="pralg_1.html#D4" target="_self">Def4</a></i>;<br/>
<b>hence </b><a NAME="E6:5_4"/>
not the <a href="unialg_1.html#U1">charact</a> of <a href="unialg_1.html#G1">UAStr</a>(# <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K4" target="_self">Opers</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> #) <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:5"/>
 <a href="unialg_1.html#G1">UAStr</a>(# <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p1">(<span class="default"><a href="pralg_1.html#K4" target="_self">Opers</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> #) is  <a href="unialg_1.html#V3">strict</a> <a href="unialg_1.html#NM2">Universal_Algebra</a>
 <b>by </b><i><a class="txt" href="pralg_1.html#E4:5"><i><font color="Green">E10</font></i></a>, <a class="txt" href="pralg_1.html#E5:5"><i><font color="Green">E11</font></i></a>, <a class="txt" href="pralg_1.html#E6:5"><i><font color="Green">E12</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/>


</div>
