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

<span class="kw">consider </span><font color="Maroon" title="c1">c</font> being  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">homogeneous</a>   <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a>   <a href="comput_1.html#V10" title="COMPUT_1:attr.10">binary</a>   <a href="aofa_000.html#V1" title="AOFA_000:attr.1">associative</a>   <a href="aofa_000.html#V2" title="AOFA_000:attr.2">unital</a>  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c2">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c3">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E4:70_1_1"/><span class="lab"><font color="Green" title="E46">A1</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="aofa_000.html#R1" title="AOFA_000:pred.1">is_a_unity_wrt</a> <font color="Maroon" title="c1">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="aofa_000.html#D3" target="_self" title="AOFA_000:def.3">Def3</a></span>;<br/>
<a NAME="E5:70_1_1"/>
 <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="comput_1.html#D26" title="COMPUT_1:def.26">COMPUT_1:def 26</a></span>;<br/>
<a NAME="E6:70_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A2</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="comput_1.html#T25" title="COMPUT_1:th.25">COMPUT_1:25</a></span>;<br/>
<a NAME="E7:70_1_1"/><span class="kw">then </span>
<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c2">a</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T157" title="FINSEQ_2:th.157">FINSEQ_2:157</a></span>;<br/>
<a NAME="E8:70_1_1"/><span class="kw">then </span>
<span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">x</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:70_1_1"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="ref" href="aofa_000.html#D1" target="_self" title="AOFA_000:def.1">Def1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">x</font> = <font color="Maroon" title="c3">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:70_1_1"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="ref" href="finseq_2.html#T158" title="FINSEQ_2:th.158">FINSEQ_2:158</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">i</font> being  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">homogeneous</a>   <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a>   <a href="comput_1.html#V11" title="COMPUT_1:attr.11">ternary</a>  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">w</font> being  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="unialg_1.html#V1" title="UNIALG_1:attr.1">homogeneous</a>   <a href="unialg_1.html#V2" title="UNIALG_1:attr.2">quasi_total</a>   <a href="comput_1.html#V10" title="COMPUT_1:attr.10">binary</a>  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>;<br/>
<a NAME="E12:70_1_1"/>
 ex <font color="Olive" title="b1">S</font> being  <a href="unialg_1.html#V3" title="UNIALG_1:attr.3">strict</a>   <a href="unialg_1.html#V8" title="UNIALG_1:attr.8">non-empty</a>   <a href="unialg_1.html#L1" title="UNIALG_1:struct.1">UAStr</a>  st <br/>( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Olive" title="b1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p4"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span> <a href="aofa_000.html#K1" title="AOFA_000:func.1">--&gt;</a> <font color="Maroon" title="c4">x</font></span>)</span>,<font color="Maroon" title="c1">c</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">w</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> &amp; <font color="Olive" title="b1">S</font> is  <a href="aofa_000.html#V3" title="AOFA_000:attr.3">with_empty-instruction</a>  &amp; <font color="Olive" title="b1">S</font> is  <a href="aofa_000.html#V4" title="AOFA_000:attr.4">with_catenation</a>  &amp; <font color="Olive" title="b1">S</font> is  <a href="aofa_000.html#V8" title="AOFA_000:attr.8">unital</a>  &amp; <font color="Olive" title="b1">S</font> is  <a href="aofa_000.html#V7" title="AOFA_000:attr.7">associative</a>  &amp; <font color="Olive" title="b1">S</font> is  <a href="aofa_000.html#V5" title="AOFA_000:attr.5">with_if-instruction</a>  &amp; <font color="Olive" title="b1">S</font> is  <a href="aofa_000.html#V6" title="AOFA_000:attr.6">with_while-instruction</a>  &amp; <font color="Olive" title="b1">S</font> is  <a href="unialg_1.html#V7" title="UNIALG_1:attr.7">quasi_total</a>  &amp; <font color="Olive" title="b1">S</font> is  <a href="unialg_1.html#V6" title="UNIALG_1:attr.6">partial</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:70_1_1"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="ref" href="aofa_000.html#T41" target="_self" title="AOFA_000:th.41">Th41</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:70_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being  <a href="unialg_1.html#V3" title="UNIALG_1:attr.3">strict</a>   <a href="unialg_1.html#V6" title="UNIALG_1:attr.6">partial</a>   <a href="unialg_1.html#V7" title="UNIALG_1:attr.7">quasi_total</a>   <a href="unialg_1.html#V8" title="UNIALG_1:attr.8">non-empty</a>   <a href="unialg_1.html#L1" title="UNIALG_1:struct.1">UAStr</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> is  <a href="aofa_000.html#V3" title="AOFA_000:attr.3">with_empty-instruction</a>  &amp; <font color="Olive">b<sub>1</sub></font> is  <a href="aofa_000.html#V4" title="AOFA_000:attr.4">with_catenation</a>  &amp; <font color="Olive">b<sub>1</sub></font> is  <a href="aofa_000.html#V5" title="AOFA_000:attr.5">with_if-instruction</a>  &amp; <font color="Olive">b<sub>1</sub></font> is  <a href="aofa_000.html#V6" title="AOFA_000:attr.6">with_while-instruction</a>  &amp; <font color="Olive">b<sub>1</sub></font> is  <a href="aofa_000.html#V8" title="AOFA_000:attr.8">unital</a>  &amp; <font color="Olive">b<sub>1</sub></font> is  <a href="aofa_000.html#V7" title="AOFA_000:attr.7">associative</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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