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<span class="kw">set </span><font color="Maroon" title="c1">A</font> =  the   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c2">a</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ;<br/>
<span class="kw">reconsider </span><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>  the   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> </span>)</span> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a>  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  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"> the   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span>, the   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ) <span class="kw">by</span> <span class="lab"><a class="ref" href="margrel1.html#T19" onmouseover="rs('margrel1/T19')" onmouseout="rh()">MARGREL1:19</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">U0</font> =  <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(#  the   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ,<span class="p1"><a href="margrel1.html#K16" title="MARGREL1:func.16">&lt;*</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="margrel1.html#K16" title="MARGREL1:func.16">*&gt;</a></span> #);<br/>
<a NAME="E2:28_1_1"/><span class="lab"><font color="Green" title="E14">A1</font></span>: 
(  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>(#  the   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ,<span class="p1"><a href="margrel1.html#K16" title="MARGREL1:func.16">&lt;*</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="margrel1.html#K16" title="MARGREL1:func.16">*&gt;</a></span> #) is  <a href="margrel1.html#V5" title="MARGREL1: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>(#  the   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ,<span class="p1"><a href="margrel1.html#K16" title="MARGREL1:func.16">&lt;*</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="margrel1.html#K16" title="MARGREL1:func.16">*&gt;</a></span> #) is  <a href="margrel1.html#V4" title="MARGREL1:attr.4">homogeneous</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="margrel1.html#T20" onmouseover="rs('margrel1/T20')" onmouseout="rh()">MARGREL1:20</a></span>;<br/>
<a NAME="E3:28_1_1"/>
 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>(#  the   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ,<span class="p1"><a href="margrel1.html#K16" title="MARGREL1:func.16">&lt;*</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="margrel1.html#K16" title="MARGREL1:func.16">*&gt;</a></span> #) is  <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="margrel1.html#T20" onmouseover="rs('margrel1/T20')" onmouseout="rh()">MARGREL1:20</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">U0</font> =  <a href="unialg_1.html#G1" title="UNIALG_1:aggr.1">UAStr</a>(#  the   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ,<span class="p1"><a href="margrel1.html#K16" title="MARGREL1:func.16">&lt;*</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="margrel1.html#K16" title="MARGREL1:func.16">*&gt;</a></span> #) as   <a href="unialg_1.html#NM1" title="UNIALG_1:NM.1">Universal_Algebra</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28_1_1"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="ref" href="unialg_1.html#D1" onmouseover="rs('unialg_1/D1')" onmouseout="rh()">UNIALG_1:def 1</a>, <a class="ref" href="unialg_1.html#D2" onmouseover="rs('unialg_1/D2')" onmouseout="rh()">UNIALG_1:def 2</a>, <a class="ref" href="unialg_1.html#D3" onmouseover="rs('unialg_1/D3')" onmouseout="rh()">UNIALG_1:def 3</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">U0</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#V1" title="UNIALG_1:attr.1">strict</a>  )</span><br/>

<a NAME="E5:28_1_1"/>
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="margrel1.html#K16" title="MARGREL1:func.16">&lt;*</a><span class="default"><font color="Maroon" title="c3">w</font></span><a href="margrel1.html#K16" title="MARGREL1:func.16">*&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="tarski.html#R2" title="TARSKI: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> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T2" onmouseover="rs('finseq_1/T2')" onmouseout="rh()">FINSEQ_1:2</a>, <a class="ref" href="finseq_1.html#T38" onmouseover="rs('finseq_1/T38')" onmouseout="rh()">FINSEQ_1:38</a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><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="partfun1.html#K8" title="PARTFUN1:func.8">.</a> 1 as   <a href="unialg_2.html#NM3" title="UNIALG_2:NM.3">operation</a> of <font color="Maroon" title="c5">U0</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E7:28_1_1"/><span class="lab"><font color="Green" title="E15">A2</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">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>  the   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> </span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" onmouseover="rs('funcop_1/T13')" onmouseout="rh()">FUNCOP_1:13</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E16">A3</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="28_1_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c6">o</font> holds <br/> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">x</font> be   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">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#K5" title="NUMBERS:func.5">0</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:28_1_1_1"/>
<font color="Maroon" title="c7">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c6">o</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K5" title="NUMBERS:func.5">0</a> </span><br/><a NAME="E2:28_1_1_1"/><span class="kw">then </span>
<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>  the   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:28_1_1"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:28_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#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E9:28_1_1"/>
 <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a>  the   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  <a href="tarski.html#R2" title="TARSKI: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>  the   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> </span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<a NAME="E10:28_1_1"/><span class="kw">then </span>
 <a href="margrel1.html#K18" title="MARGREL1:func.18">arity</a> <font color="Maroon" title="c6">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:28_1_1"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="txt" href="#E8:28_1_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="ref" href="margrel1.html#D25" onmouseover="rs('margrel1/D25')" onmouseout="rh()">MARGREL1:def 25</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:28_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#V1" title="UNIALG_1:attr.1">strict</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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