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

<b>consider </b><font color="Maroon" title="c1">R</font> being  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="realset2.html#V3" title="REALSET2:attr.3">trivial</a> <a href="orders_2.html#V2" title="ORDERS_2:attr.2">reflexive</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c2">x</font> = 0, <font color="Maroon" title="c3">y</font> = 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">0,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">n</font> = 0,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c3">y</font>,<font color="Maroon" title="c2">x</font> as   <a href="binop_1.html#NM1" title="BINOP_1:NM.1">UnOp</a> of <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">0,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> ;<br/>
<a NAME="E4:43_1_1"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">0,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T12" title="ZFMISC_1:th.12">ZFMISC_1:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">A</font> = <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">0</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">0,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <b>by </b><i><a class="ref" href="finsub_1.html#D5" title="FINSUB_1:def.5">FINSUB_1:def 5</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">f</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <a href="borsuk_1.html#K1" title="BORSUK_1:func.1">--&gt;</a> <font color="Maroon" title="c5">A</font>;<br/>
<b>set </b><font color="Maroon" title="c7">T</font> =  <a href="abcmiz_0.html#G2" title="ABCMIZ_0:aggr.2">TA-structure</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">0,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c4">n</font>,<span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <a href="borsuk_1.html#K1" title="BORSUK_1:func.1">--&gt;</a> <font color="Maroon" title="c5">A</font></span>)</span> #);<br/>
<a NAME="E6:43_1_1"/>
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="abcmiz_0.html#G2" title="ABCMIZ_0:aggr.2">TA-structure</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">0,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c4">n</font>,<span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <a href="borsuk_1.html#K1" title="BORSUK_1:func.1">--&gt;</a> <font color="Maroon" title="c5">A</font></span>)</span> #),the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <a href="abcmiz_0.html#G2" title="ABCMIZ_0:aggr.2">TA-structure</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">0,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c4">n</font>,<span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <a href="borsuk_1.html#K1" title="BORSUK_1:func.1">--&gt;</a> <font color="Maroon" title="c5">A</font></span>)</span> #) #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font> #)
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">T</font> =  <a href="abcmiz_0.html#G2" title="ABCMIZ_0:aggr.2">TA-structure</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">0,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c4">n</font>,<span class="p1">(<span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <a href="borsuk_1.html#K1" title="BORSUK_1:func.1">--&gt;</a> <font color="Maroon" title="c5">A</font></span>)</span> #) as  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="realset2.html#V3" title="REALSET2:attr.3">trivial</a> <a href="orders_2.html#V2" title="ORDERS_2:attr.2">reflexive</a> <a href="abcmiz_0.html#V7" title="ABCMIZ_0:attr.7">strict</a>  <a href="abcmiz_0.html#L2" title="ABCMIZ_0:struct.2">TA-structure</a>  <b>by </b><i><a class="ref" href="tex_2.html#T4" title="TEX_2:th.4">TEX_2:4</a>, <a class="ref" href="waybel_8.html#T12" title="WAYBEL_8:th.12">WAYBEL_8:12</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c8">T</font>
;<br/>

<b>thus </b><a NAME="E8:43_1_1"/>
not <font color="Maroon" title="c8">T</font> is <a href="abcmiz_0.html#V4" title="ABCMIZ_0:attr.4">void</a>
 ;<br/>

<b>thus </b><a NAME="E9:43_1_1"/>
<font color="Maroon" title="c8">T</font> is <a href="abcmiz_0.html#V2" title="ABCMIZ_0:attr.2">Mizar-widening-like</a>
 ;<br/>

<a NAME="E10:43_1_1"/><i><font color="Green" title="E21">A1</font></i>: 
( 0 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp; <font color="Maroon" title="c4">n</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font> &amp; <font color="Maroon" title="c4">n</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> )
 
<b>by </b><i><a class="ref" href="funct_4.html#T66" title="FUNCT_4:th.66">FUNCT_4:66</a></i>;<br/>
<b>hence </b><a NAME="E11:43_1_1"/>
( <font color="Maroon" title="c8">T</font> is <a href="abcmiz_0.html#V5" title="ABCMIZ_0:attr.5">involutive</a> &amp; <font color="Maroon" title="c8">T</font> is <a href="abcmiz_0.html#V6" title="ABCMIZ_0:attr.6">without_fixpoints</a> )
 <b>by </b><i><a class="ref" href="abcmiz_0.html#T4" target="_self" title="ABCMIZ_0:th.4">Th4</a></i>;<br/>

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="abcmiz_0.html#D9" target="_self" title="ABCMIZ_0:def.9">ABCMIZ_0:def 9</a>
<div class="add"><b>let </b><font color="Maroon" title="c9">t</font> be   <a href="abcmiz_0.html#NM1" title="ABCMIZ_0:NM.1">type</a> of <font color="Maroon" title="c8">T</font>;<br/><b>let </b><font color="Maroon" title="c10">a</font> be   <a href="abcmiz_0.html#NM2" title="ABCMIZ_0:NM.2">adjective</a> of <font color="Maroon" title="c8">T</font>;<br/><a NAME="E1:43_1_1_1"/><i><font color="Green" title="E22">A2</font></i>: 
( ( <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) &amp;  <a href="abcmiz_0.html#K1" title="ABCMIZ_0:func.1">non-</a> <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">a</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><a NAME="E2:43_1_1_1"/>
 <a href="abcmiz_0.html#K2" title="ABCMIZ_0:func.2">adjs</a> <font color="Maroon" title="c9">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A</font>
 <b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/><b>hence </b><a NAME="E3:43_1_1_1"/>
( <font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="abcmiz_0.html#K2" title="ABCMIZ_0:func.2">adjs</a> <font color="Maroon" title="c9">t</font> implies  not  <a href="abcmiz_0.html#K1" title="ABCMIZ_0:func.1">non-</a> <font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="abcmiz_0.html#K2" title="ABCMIZ_0:func.2">adjs</a> <font color="Maroon" title="c9">t</font> )
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E10:43_1_1"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="abcmiz_0.html#E1:43_1_1_1"><i><font color="Green" title="E22">A2</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div>
<b>end;</b></div>

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="abcmiz_0.html#D11" target="_self" title="ABCMIZ_0:def.11">ABCMIZ_0:def 11</a>
<div class="add"><b>let </b><font color="Maroon" title="c9">t1</font>, <font color="Maroon" title="c10">t2</font> be   <a href="abcmiz_0.html#NM1" title="ABCMIZ_0:NM.1">type</a> of <font color="Maroon" title="c8">T</font>;<br/><a NAME="E1:43_1_1_2"/>
(  <a href="abcmiz_0.html#K2" title="ABCMIZ_0:func.2">adjs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">t1</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">t2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A</font> &amp;  <a href="abcmiz_0.html#K2" title="ABCMIZ_0:func.2">adjs</a> <font color="Maroon" title="c9">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A</font> &amp;  <a href="abcmiz_0.html#K2" title="ABCMIZ_0:func.2">adjs</a> <font color="Maroon" title="c10">t2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A</font> )
 <b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/><b>hence </b><a NAME="E2:43_1_1_2"/>
 <a href="abcmiz_0.html#K2" title="ABCMIZ_0:func.2">adjs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">t1</font> <a href="lattice3.html#K13" title="LATTICE3:func.13">"\/"</a> <font color="Maroon" title="c10">t2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K2" title="ABCMIZ_0:func.2">adjs</a> <font color="Maroon" title="c9">t1</font></span>)</span> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K2" title="ABCMIZ_0:func.2">adjs</a> <font color="Maroon" title="c10">t2</font></span>)</span>
 ;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon" title="c9">a</font> be   <a href="abcmiz_0.html#NM2" title="ABCMIZ_0:NM.2">adjective</a> of <font color="Maroon" title="c8">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="abcmiz_0.html#D14" target="_self" title="ABCMIZ_0:def.14">ABCMIZ_0:def 14</a><br/>

<a NAME="E14:43_1_1"/><i><font color="Green" title="E22">A3</font></i>: 
( ( <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) &amp;  <a href="abcmiz_0.html#K1" title="ABCMIZ_0:func.1">non-</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">n</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">a</font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">t</font> being   <a href="abcmiz_0.html#NM1" title="ABCMIZ_0:NM.1">type</a> of <font color="Maroon" title="c8">T</font>;<br/>
<a NAME="E16:43_1_1"/>
 <a href="abcmiz_0.html#K2" title="ABCMIZ_0:func.2">adjs</a> <font color="Maroon" title="c10">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">A</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E17:43_1_1"/><b>then </b>
( <font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="abcmiz_0.html#K2" title="ABCMIZ_0:func.2">adjs</a> <font color="Maroon" title="c10">t</font> or  <a href="abcmiz_0.html#K1" title="ABCMIZ_0:func.1">non-</a> <font color="Maroon" title="c9">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="abcmiz_0.html#K2" title="ABCMIZ_0:func.2">adjs</a> <font color="Maroon" title="c10">t</font> )
 
<b>by </b><i><a class="txt" href="abcmiz_0.html#E10:43_1_1"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="abcmiz_0.html#E14:43_1_1"><i><font color="Green" title="E22">A3</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E18:43_1_1"/><b>then </b>
( <font color="Maroon" title="c10">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="abcmiz_0.html#K3" title="ABCMIZ_0:func.3">types</a> <font color="Maroon" title="c9">a</font> or <font color="Maroon" title="c10">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="abcmiz_0.html#K3" title="ABCMIZ_0:func.3">types</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K1" title="ABCMIZ_0:func.1">non-</a> <font color="Maroon" title="c9">a</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="abcmiz_0.html#T13" target="_self" title="ABCMIZ_0:th.13">Th13</a></i>;<br/>
<b>hence </b><a NAME="E19:43_1_1"/>
not <span class="p1">(<span class="default"><a href="abcmiz_0.html#K3" title="ABCMIZ_0:func.3">types</a> <font color="Maroon" title="c9">a</font></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K3" title="ABCMIZ_0:func.3">types</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K1" title="ABCMIZ_0:func.1">non-</a> <font color="Maroon" title="c9">a</font></span>)</span></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>


</div>
