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<span class="kw">let </span><font color="Maroon" title="c1">S</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="struct_0.html#L1" title="STRUCT_0:struct.1">1-sorted</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">N</font> being  <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">S</font><br/> for <font color="Olive" title="b2">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">N</font> <a href="waybel_0.html#R2" title="WAYBEL_0:pred.2">is_often_in</a> <font color="Olive" title="b2">X</font> holds <br/><font color="Olive" title="b1">N</font> <a href="yellow_6.html#K5" title="YELLOW_6:func.5">"</a> <font color="Olive" title="b2">X</font> is    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Olive" title="b1">N</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">N</font> be   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">X</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Maroon" title="c2">N</font> <a href="waybel_0.html#R2" title="WAYBEL_0:pred.2">is_often_in</a> <font color="Olive" title="b1">X</font> holds <br/><font color="Maroon" title="c2">N</font> <a href="yellow_6.html#K5" title="YELLOW_6:func.5">"</a> <font color="Olive" title="b1">X</font> is    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c2">N</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">N</font> <a href="waybel_0.html#R2" title="WAYBEL_0:pred.2">is_often_in</a> <font color="Maroon" title="c3">X</font> implies <font color="Maroon" title="c2">N</font> <a href="yellow_6.html#K5" title="YELLOW_6:func.5">"</a> <font color="Maroon" title="c3">X</font> is    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c2">N</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:72"/><span class="lab"><font color="Green" title="E33">A1</font></span>: 
<font color="Maroon" title="c2">N</font> <a href="waybel_0.html#R2" title="WAYBEL_0:pred.2">is_often_in</a> <font color="Maroon" title="c3">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">N</font> <a href="yellow_6.html#K5" title="YELLOW_6:func.5">"</a> <font color="Maroon" title="c3">X</font> is    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c2">N</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">M</font> = <font color="Maroon" title="c2">N</font> <a href="yellow_6.html#K5" title="YELLOW_6:func.5">"</a> <font color="Maroon" title="c3">X</font> as   <a href="waybel_0.html#NM2" title="WAYBEL_0:NM.2">net</a> of <font color="Maroon" title="c1">S</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_6.html#T30" target="_self" title="YELLOW_6:th.30">Th30</a></span>;<br/>
<a NAME="E3:72"/>
<font color="Maroon" title="c4">M</font> is    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c2">N</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c5">f</font> =  <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c4">M</font>;<br/>
<a NAME="E1:72_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">N</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_6.html#T19" target="_self" title="YELLOW_6:th.19">Th19</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">f</font> =  <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <font color="Maroon" title="c4">M</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c4">M</font>,<font color="Maroon" title="c2">N</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T9" title="FUNCT_2:th.9">FUNCT_2:9</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c6">f</font>
; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="yellow_6.html#D12" target="_self" title="YELLOW_6:def.12">YELLOW_6:def 12</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c4">M</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c2">N</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">f</font> &amp; ( for <font color="Olive" title="b1">m</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font>  ex <font color="Olive" title="b2">n</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">M</font> st <br/>for <font color="Olive" title="b3">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">M</font>  st <font color="Olive" title="b2">n</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b3">p</font> holds <br/><font color="Olive" title="b1">m</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">f</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b3">p</font> ) )</span><br/>

<a NAME="E3:72_1"/>
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c4">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c2">N</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_6.html#D8" target="_self" title="YELLOW_6:def.8">Def8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:72_1"/>
the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c4">M</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c2">N</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T94" title="RELAT_1:th.94">RELAT_1:94</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">m</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font>  ex <font color="Olive" title="b2">n</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">M</font> st <br/>for <font color="Olive" title="b3">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">M</font>  st <font color="Olive" title="b2">n</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b3">p</font> holds <br/><font color="Olive" title="b1">m</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">f</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b3">p</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c7">m</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">n</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">M</font> st <br/>for <font color="Olive" title="b2">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">M</font>  st <font color="Olive" title="b1">n</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b2">p</font> holds <br/><font color="Maroon" title="c7">m</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">f</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b2">p</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c8">j</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">N</font><span class="kw"> such that </span><br/><a NAME="E6:72_1"/><span class="lab"><font color="Green" title="E34">A2</font></span>: 
<font color="Maroon" title="c7">m</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c8">j</font>
 <span class="kw">and </span><br/><a NAME="E7:72_1"/><span class="lab"><font color="Green" title="E35">A3</font></span>: 
<font color="Maroon" title="c2">N</font> <a href="waybel_0.html#K2" title="WAYBEL_0:func.2">.</a> <font color="Maroon" title="c8">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:72"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="ref" href="waybel_0.html#D12" title="WAYBEL_0:def.12">WAYBEL_0:def 12</a></span>;<br/>
<a NAME="E8:72_1"/>
<font color="Maroon" title="c8">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="waybel_0.html#U1" title="WAYBEL_0:sel.1">mapping</a> of <font color="Maroon" title="c2">N</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c3">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:72_1"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="ref" href="funct_2.html#T46" title="FUNCT_2:th.46">FUNCT_2:46</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">n</font> = <font color="Maroon" title="c8">j</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">M</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_6.html#D13" target="_self" title="YELLOW_6:def.13">Def13</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c9">n</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">M</font>  st <font color="Maroon" title="c9">n</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Olive" title="b1">p</font> holds <br/><font color="Maroon" title="c7">m</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">f</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Olive" title="b1">p</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c10">p</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">n</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c10">p</font> implies <font color="Maroon" title="c7">m</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">f</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Maroon" title="c10">p</font> )</span><br/>

<span class="kw">assume </span><a NAME="E10:72_1"/><span class="lab"><font color="Green" title="E36">A4</font></span>: 
<font color="Maroon" title="c9">n</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c10">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">m</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">f</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Maroon" title="c10">p</font></span><br/>

<a NAME="E11:72_1"/>
<font color="Maroon" title="c6">f</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Maroon" title="c10">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></span>;<br/>
<a NAME="E12:72_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">j</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">f</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Maroon" title="c10">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:72_1"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="ref" href="yellow_6.html#T20" target="_self" title="YELLOW_6:th.20">Th20</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:72_1"/>
<font color="Maroon" title="c7">m</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a> <font color="Maroon" title="c6">f</font> <a href="yellow_6.html#K2" title="YELLOW_6:func.2">.</a> <font color="Maroon" title="c10">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:72_1"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="ref" href="yellow_0.html#D2" title="YELLOW_0:def.2">YELLOW_0:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:72"/>
<font color="Maroon" title="c2">N</font> <a href="yellow_6.html#K5" title="YELLOW_6:func.5">"</a> <font color="Maroon" title="c3">X</font> is    <a href="yellow_6.html#M2" title="YELLOW_6:mode.2">subnet</a> of <font color="Maroon" title="c2">N</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
