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<span class="kw">let </span><font color="Maroon" title="c1">S1</font> be   <a href="osalg_1.html#NM3" title="OSALG_1:NM.3">OrderSortedSign</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">  for <font color="Olive" title="b1">OU0</font> being   <a href="osalg_1.html#NM5" title="OSALG_1:NM.5">OSAlgebra</a> of <font color="Maroon" title="c1">S1</font><br/>  for <font color="Olive" title="b2">A</font> being    <a href="osalg_2.html#M2" title="OSALG_2:mode.2">OSSubset</a> of <font color="Olive" title="b1">OU0</font><br/>  for <font color="Olive" title="b3">s1</font>, <font color="Olive" title="b4">s2</font> being   <a href="osalg_1.html#NM1" title="OSALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S1</font>  st <font color="Olive" title="b3">s1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b4">s2</font> holds <br/> <a href="osalg_2.html#K8" title="OSALG_2:func.8">OSSubSort</a> (<font color="Olive" title="b2">A</font>,<font color="Olive" title="b4">s2</font>) <a href="setfam_1.html#R2" title="SETFAM_1:pred.2">is_coarser_than</a>  <a href="osalg_2.html#K8" title="OSALG_2:func.8">OSSubSort</a> (<font color="Olive" title="b2">A</font>,<font color="Olive" title="b3">s1</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c2">OU0</font> be   <a href="osalg_1.html#NM5" title="OSALG_1:NM.5">OSAlgebra</a> of <font color="Maroon" title="c1">S1</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">  for <font color="Olive" title="b1">A</font> being    <a href="osalg_2.html#M2" title="OSALG_2:mode.2">OSSubset</a> of <font color="Maroon" title="c2">OU0</font><br/>  for <font color="Olive" title="b2">s1</font>, <font color="Olive" title="b3">s2</font> being   <a href="osalg_1.html#NM1" title="OSALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S1</font>  st <font color="Olive" title="b2">s1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b3">s2</font> holds <br/> <a href="osalg_2.html#K8" title="OSALG_2:func.8">OSSubSort</a> (<font color="Olive" title="b1">A</font>,<font color="Olive" title="b3">s2</font>) <a href="setfam_1.html#R2" title="SETFAM_1:pred.2">is_coarser_than</a>  <a href="osalg_2.html#K8" title="OSALG_2:func.8">OSSubSort</a> (<font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">s1</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c3">A</font> be    <a href="osalg_2.html#M2" title="OSALG_2:mode.2">OSSubset</a> of <font color="Maroon" title="c2">OU0</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">  for <font color="Olive" title="b1">s1</font>, <font color="Olive" title="b2">s2</font> being   <a href="osalg_1.html#NM1" title="OSALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S1</font>  st <font color="Olive" title="b1">s1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b2">s2</font> holds <br/> <a href="osalg_2.html#K8" title="OSALG_2:func.8">OSSubSort</a> (<font color="Maroon" title="c3">A</font>,<font color="Olive" title="b2">s2</font>) <a href="setfam_1.html#R2" title="SETFAM_1:pred.2">is_coarser_than</a>  <a href="osalg_2.html#K8" title="OSALG_2:func.8">OSSubSort</a> (<font color="Maroon" title="c3">A</font>,<font color="Olive" title="b1">s1</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c4">s1</font>, <font color="Maroon" title="c5">s2</font> be   <a href="osalg_1.html#NM1" title="OSALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S1</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="c4">s1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">s2</font> implies  <a href="osalg_2.html#K8" title="OSALG_2:func.8">OSSubSort</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c5">s2</font>) <a href="setfam_1.html#R2" title="SETFAM_1:pred.2">is_coarser_than</a>  <a href="osalg_2.html#K8" title="OSALG_2:func.8">OSSubSort</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">s1</font>) )</span><br/>







<span class="kw">assume </span><a NAME="E1:41"/><span class="lab"><font color="Green" title="E24">A1</font></span>: 
<font color="Maroon" title="c4">s1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c5">s2</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="osalg_2.html#K8" title="OSALG_2:func.8">OSSubSort</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c5">s2</font>) <a href="setfam_1.html#R2" title="SETFAM_1:pred.2">is_coarser_than</a>  <a href="osalg_2.html#K8" title="OSALG_2:func.8">OSSubSort</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">s1</font>)</span><br/>

<a NAME="E2:41"/>
 for <font color="Olive" title="b1">Y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">Y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="osalg_2.html#K8" title="OSALG_2:func.8">OSSubSort</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c5">s2</font>) holds <br/> ex <font color="Olive" title="b2">X</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>( <font color="Olive" title="b2">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="osalg_2.html#K8" title="OSALG_2:func.8">OSSubSort</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">s1</font>) &amp; <font color="Olive" title="b2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">Y</font> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="41_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</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="c6">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="osalg_2.html#K8" title="OSALG_2:func.8">OSSubSort</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c5">s2</font>) implies  ex <font color="Olive" title="b1">X</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>( <font color="Olive" title="b1">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="osalg_2.html#K8" title="OSALG_2:func.8">OSSubSort</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">s1</font>) &amp; <font color="Olive" title="b1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">x</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:41_1"/>
<font color="Maroon" title="c6">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="osalg_2.html#K8" title="OSALG_2:func.8">OSSubSort</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c5">s2</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">  ex <font color="Olive" title="b1">X</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>( <font color="Olive" title="b1">X</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="osalg_2.html#K8" title="OSALG_2:func.8">OSSubSort</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">s1</font>) &amp; <font color="Olive" title="b1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">x</font> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">B</font> being    <a href="osalg_2.html#M2" title="OSALG_2:mode.2">OSSubset</a> of <font color="Maroon" title="c2">OU0</font><span class="kw"> such that </span><br/><a NAME="E3:41_1"/><span class="lab"><font color="Green" title="E25">A2</font></span>: 
( <font color="Maroon" title="c7">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="osalg_2.html#K5" title="OSALG_2:func.5">OSSubSort</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s2</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="osalg_2.html#D10" target="_self" onmouseover="rs('osalg_2/D10')" onmouseout="rh()">Def10</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c7">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">s1</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="c7">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">s1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="osalg_2.html#K8" title="OSALG_2:func.8">OSSubSort</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">s1</font>) &amp; <font color="Maroon" title="c7">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">x</font> )</span><br/>

<a NAME="E4:41_1"/>
<font color="Maroon" title="c7">B</font> is   <a href="osalg_1.html#NM4" title="OSALG_1:NM.4">OrderSortedSet</a> of <font color="Maroon" title="c1">S1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="osalg_2.html#D2" target="_self" onmouseover="rs('osalg_2/D2')" onmouseout="rh()">Def2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:41_1"/>
( <font color="Maroon" title="c7">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">s1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="osalg_2.html#K8" title="OSALG_2:func.8">OSSubSort</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">s1</font>) &amp; <font color="Maroon" title="c7">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">x</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E3:41_1"><span class="lab"><font color="Green" title="E25">A2</font></span></a>, <a class="ref" href="osalg_2.html#D10" target="_self" onmouseover="rs('osalg_2/D10')" onmouseout="rh()">Def10</a>, <a class="ref" href="osalg_1.html#D16" onmouseover="rs('osalg_1/D16')" onmouseout="rh()">OSALG_1:def 16</a></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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:41"/>
 <a href="osalg_2.html#K8" title="OSALG_2:func.8">OSSubSort</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c5">s2</font>) <a href="setfam_1.html#R2" title="SETFAM_1:pred.2">is_coarser_than</a>  <a href="osalg_2.html#K8" title="OSALG_2:func.8">OSSubSort</a> (<font color="Maroon" title="c3">A</font>,<font color="Maroon" title="c4">s1</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="setfam_1.html#D3" onmouseover="rs('setfam_1/D3')" onmouseout="rh()">SETFAM_1:def 3</a></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"> verum</span><br/>


</div>
