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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> holds   <a href="osalg_2.html#K5" title="OSALG_2:func.5">OSSubSort</a> <font color="Olive" title="b2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="osalg_2.html#K6" title="OSALG_2:func.6">OSSubSort</a> <font color="Olive" title="b1">OU0</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> holds   <a href="osalg_2.html#K5" title="OSALG_2:func.5">OSSubSort</a> <font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="osalg_2.html#K6" title="OSALG_2:func.6">OSSubSort</a> <font color="Maroon" title="c2">OU0</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">  <a href="osalg_2.html#K5" title="OSALG_2:func.5">OSSubSort</a> <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="osalg_2.html#K6" title="OSALG_2:func.6">OSSubSort</a> <font color="Maroon" title="c2">OU0</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osalg_2.html#K5" title="OSALG_2:func.5">OSSubSort</a> <font color="Maroon" title="c3">A</font> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osalg_2.html#K6" title="OSALG_2:func.6">OSSubSort</a> <font color="Maroon" title="c2">OU0</font> )</span><br/>







<span class="kw">assume </span><a NAME="E1:35"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osalg_2.html#K5" title="OSALG_2:func.5">OSSubSort</a> <font color="Maroon" title="c3">A</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osalg_2.html#K6" title="OSALG_2:func.6">OSSubSort</a> <font color="Maroon" title="c2">OU0</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">x1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_2.html#K5" title="MSUALG_2:func.5">SubSort</a> <font color="Maroon" title="c3">A</font><span class="kw"> such that </span><br/><a NAME="E3:35"/><span class="lab"><font color="Green" title="E20">A1</font></span>: 
<font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font>
 <span class="kw">and </span><br/><a NAME="E4:35"/><span class="lab"><font color="Green" title="E21">A2</font></span>: 
<font color="Maroon" title="c5">x1</font> is   <a href="osalg_1.html#NM4" title="OSALG_1:NM.4">OrderSortedSet</a> of <font color="Maroon" title="c1">S1</font>
 ;<br/>
<a NAME="E5:35"/>
( <font color="Maroon" title="c5">x1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="msualg_2.html#K5" title="MSUALG_2:func.5">SubSort</a> <font color="Maroon" title="c3">A</font> &amp;  <a href="msualg_2.html#K5" title="MSUALG_2:func.5">SubSort</a> <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="msualg_2.html#K6" title="MSUALG_2:func.6">SubSort</a> <font color="Maroon" title="c2">OU0</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_2.html#T39" onmouseover="rs('msualg_2/T39')" onmouseout="rh()">MSUALG_2:39</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">x2</font> = <font color="Maroon" title="c5">x1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_2.html#K6" title="MSUALG_2:func.6">SubSort</a> <font color="Maroon" title="c2">OU0</font> ;<br/>
<a NAME="E7:35"/>
<font color="Maroon" title="c6">x2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_2.html#K6" title="MSUALG_2:func.6">SubSort</a> <font color="Maroon" title="c2">OU0</font> : <font color="Olive" title="b1">y</font> is   <a href="osalg_1.html#NM4" title="OSALG_1:NM.4">OrderSortedSet</a> of <font color="Maroon" title="c1">S1</font> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35"><span class="lab"><font color="Green" title="E21">A2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:35"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osalg_2.html#K6" title="OSALG_2:func.6">OSSubSort</a> <font color="Maroon" title="c2">OU0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:35"><span class="lab"><font color="Green" title="E20">A1</font></span></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/>


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