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<b>consider </b><font color="Maroon" title="c1">A</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>
<b>consider </b><font color="Maroon" title="c2">a</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">A</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c3">w</font> = <span class="p1">(<span class="default"><a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <font color="Maroon" title="c1">A</font></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c2">a</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">A</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,<font color="Maroon" title="c1">A</font> <b>by </b><i><a class="ref" href="unialg_1.html#T3" title="UNIALG_1:th.3">UNIALG_1:3</a></i>;<br/>
<b>set </b><font color="Maroon" title="c4">U0</font> =  <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1"><a href="unialg_1.html#K1" title="UNIALG_1:func.1">&lt;*</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="unialg_1.html#K1" title="UNIALG_1:func.1">*&gt;</a></span> #);<br/>
<a NAME="E4:33_1_1"/><i><font color="Green" title="E20">A1</font></i>: 
( the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1"><a href="unialg_1.html#K1" title="UNIALG_1:func.1">&lt;*</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="unialg_1.html#K1" title="UNIALG_1:func.1">*&gt;</a></span> #) is <a href="unialg_1.html#V5" title="UNIALG_1:attr.5">quasi_total</a> &amp; the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1"><a href="unialg_1.html#K1" title="UNIALG_1:func.1">&lt;*</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="unialg_1.html#K1" title="UNIALG_1:func.1">*&gt;</a></span> #) is <a href="unialg_1.html#V4" title="UNIALG_1:attr.4">homogeneous</a> &amp; the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1"><a href="unialg_1.html#K1" title="UNIALG_1:func.1">&lt;*</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="unialg_1.html#K1" title="UNIALG_1:func.1">*&gt;</a></span> #) is <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> )
 
<b>by </b><i><a class="ref" href="unialg_1.html#T4" title="UNIALG_1:th.4">UNIALG_1:4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">U0</font> =  <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(# <font color="Maroon" title="c1">A</font>,<span class="p1"><a href="unialg_1.html#K1" title="UNIALG_1:func.1">&lt;*</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="unialg_1.html#K1" title="UNIALG_1:func.1">*&gt;</a></span> #) as   <a href="unialg_1.html#NM2" title="UNIALG_1:NM.2">Universal_Algebra</a> <b>by </b><i><a class="txt" href="#E4:33_1_1"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="unialg_1.html#D7" title="UNIALG_1:def.7">UNIALG_1:def 7</a>, <a class="ref" href="unialg_1.html#D8" title="UNIALG_1:def.8">UNIALG_1:def 8</a>, <a class="ref" href="unialg_1.html#D9" title="UNIALG_1:def.9">UNIALG_1:def 9</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c5">U0</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">U0</font> is <a href="unialg_2.html#V2" title="UNIALG_2:attr.2">with_const_op</a> &amp; <font color="Maroon" title="c5">U0</font> is <a href="unialg_1.html#V3" title="UNIALG_1:attr.3">strict</a> )</span><br/>

<a NAME="E6:33_1_1"/>
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="unialg_1.html#K1" title="UNIALG_1:func.1">&lt;*</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="unialg_1.html#K1" title="UNIALG_1:func.1">*&gt;</a></span> <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">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#T55" title="FINSEQ_1:th.55">FINSEQ_1:55</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">o</font> = the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <font color="Maroon" title="c5">U0</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 as   <a href="unialg_2.html#NM5" title="UNIALG_2:NM.5">operation</a> of <font color="Maroon" title="c5">U0</font> <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E8:33_1_1"/>
<font color="Maroon" title="c6">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">w</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>;<br/>
<a NAME="E9:33_1_1"/><b>then </b><i><font color="Green" title="E21">A3</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">o</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"><span class="p2">(<span class="default"><a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <font color="Maroon" title="c1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <font color="Maroon" title="c1">A</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">x</font> be   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">o</font> implies  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/><b>assume </b><a NAME="E1:33_1_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">o</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><a NAME="E2:33_1_1_1"/><b>then </b>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <font color="Maroon" title="c1">A</font>
 <b>by </b><i><a class="txt" href="#E9:33_1_1"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#T32" title="FINSEQ_1:th.32">FINSEQ_1:32</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E11:33_1_1"/><b>then </b>
 <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c6">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="#E9:33_1_1"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="unialg_1.html#D10" title="UNIALG_1:def.10">UNIALG_1:def 10</a></i>;<br/>
<b>hence </b><a NAME="E12:33_1_1"/>
( <font color="Maroon" title="c5">U0</font> is <a href="unialg_2.html#V2" title="UNIALG_2:attr.2">with_const_op</a> &amp; <font color="Maroon" title="c5">U0</font> is <a href="unialg_1.html#V3" title="UNIALG_1:attr.3">strict</a> )
 <b>by </b><i><a class="ref" href="unialg_2.html#D12" target="_self" title="UNIALG_2:def.12">Def12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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