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<span class="kw">let </span><font color="Maroon" title="c3">W1</font>, <font color="Maroon" title="c4">W2</font> be   <a href="osalg_1.html#NM4" title="OSALG_1:NM.4">OrderSortedSet</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">s</font> being   <a href="osalg_1.html#NM1" title="OSALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S1</font> holds  <font color="Maroon" title="c3">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">s1</font></span>)</span> where <font color="Olive" title="b2">s1</font> is   <a href="osalg_1.html#NM1" title="OSALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S1</font> : <font color="Olive" title="b2">s1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">s</font> </span> } </span>  ) &amp; (  for <font color="Olive" title="b1">s</font> being   <a href="osalg_1.html#NM1" title="OSALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S1</font> holds  <font color="Maroon" title="c4">W2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">s1</font></span>)</span> where <font color="Olive" title="b2">s1</font> is   <a href="osalg_1.html#NM1" title="OSALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S1</font> : <font color="Olive" title="b2">s1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">s</font> </span> } </span>  ) implies <font color="Maroon" title="c3">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:18_1_2"/><span class="lab"><font color="Green" title="E9">A6</font></span>: 
 for <font color="Olive" title="b1">s</font> being   <a href="osalg_1.html#NM1" title="OSALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S1</font> holds  <font color="Maroon" title="c3">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">s1</font></span>)</span> where <font color="Olive" title="b2">s1</font> is   <a href="osalg_1.html#NM1" title="OSALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S1</font> : <font color="Olive" title="b2">s1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">s</font> </span> } </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"> (  ex <font color="Olive" title="b1">s</font> being   <a href="osalg_1.html#NM1" title="OSALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S1</font> st  not <font color="Maroon" title="c4">W2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">s1</font></span>)</span> where <font color="Olive" title="b2">s1</font> is   <a href="osalg_1.html#NM1" title="OSALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S1</font> : <font color="Olive" title="b2">s1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">s</font> </span> } </span>  or <font color="Maroon" title="c3">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:18_1_2"/><span class="lab"><font color="Green" title="E10">A7</font></span>: 
 for <font color="Olive" title="b1">s</font> being   <a href="osalg_1.html#NM1" title="OSALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S1</font> holds  <font color="Maroon" title="c4">W2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">s1</font></span>)</span> where <font color="Olive" title="b2">s1</font> is   <a href="osalg_1.html#NM1" title="OSALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S1</font> : <font color="Olive" title="b2">s1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">s</font> </span> } </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"> <font color="Maroon" title="c3">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W2</font></span><br/>

<a NAME="E3:18_1_2"/>
 for <font color="Olive" title="b1">s</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S1</font> holds <br/><font color="Maroon" title="c3">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="18_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">s</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c5">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S1</font> implies <font color="Maroon" title="c3">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:18_1_2_1"/>
<font color="Maroon" title="c5">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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="c3">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">t</font> = <font color="Maroon" title="c5">s</font> as   <a href="osalg_1.html#NM1" title="OSALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S1</font> ;<br/>
<font color="Maroon" title="c3">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s</font> = 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c2">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s1</font></span>)</span> where <font color="Olive" title="b1">s1</font> is   <a href="osalg_1.html#NM1" title="OSALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S1</font> : <font color="Olive" title="b1">s1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">t</font> </span> } </span> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:18_1_2"><span class="lab"><font color="Green" title="E9">A6</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c4">W2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:18_1_2"><span class="lab"><font color="Green" title="E10">A7</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E4:18_1_2_1"/>
<font color="Maroon" title="c3">W1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">s</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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:18_1_2"/>
<font color="Maroon" title="c3">W1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">W2</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"> verum</span><br/>


</div>
