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<b>let </b><font color="Maroon" title="c1">S</font> be  <a href="osalg_1.html#V8" title="OSALG_1:attr.8">monotone</a> <a href="osalg_1.html#V10" title="OSALG_1:attr.10">regular</a> <a href="osalg_4.html#V2" title="OSALG_4:attr.2">locally_directed</a> <a href="osalg_1.html#NM3" title="OSALG_1:NM.3">OrderSortedSign</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">X</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">V5</a> <a href="struct_0.html#NM8" title="STRUCT_0:NM.8">ManySortedSet</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>  holds <br/> ( <font color="Olive" title="b2">x</font> is    <a href="osafree.html#M2" title="OSAFREE:mode.2">MinTerm</a> of <font color="Maroon" title="c1">S</font>,<font color="Olive" title="b1">X</font> iff <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osafree.html#K30" title="OSAFREE:func.30">MinTerms</a> <font color="Olive" title="b1">X</font> )</span><br/><b>let </b><font color="Maroon" title="c2">X</font> be  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V5</a> <a href="struct_0.html#NM8" title="STRUCT_0:NM.8">ManySortedSet</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> is    <a href="osafree.html#M2" title="OSAFREE:mode.2">MinTerm</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">X</font> iff <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osafree.html#K30" title="OSAFREE:func.30">MinTerms</a> <font color="Maroon" title="c2">X</font> )</span><br/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> is    <a href="osafree.html#M2" title="OSAFREE:mode.2">MinTerm</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">X</font> iff <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osafree.html#K30" title="OSAFREE:func.30">MinTerms</a> <font color="Maroon" title="c2">X</font> )</span><br/>





<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osafree.html#K30" title="OSAFREE:func.30">MinTerms</a> <font color="Maroon" title="c2">X</font> implies <font color="Maroon" title="c3">x</font> is    <a href="osafree.html#M2" title="OSAFREE:mode.2">MinTerm</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">X</font> )</span>
<div class="add"><b>assume </b><a NAME="E1:95_1"/>
<font color="Maroon" title="c3">x</font> is    <a href="osafree.html#M2" title="OSAFREE:mode.2">MinTerm</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osafree.html#K30" title="OSAFREE:func.30">MinTerms</a> <font color="Maroon" title="c2">X</font></span><br/><b>then reconsider </b><font color="Maroon" title="c4">t</font> = <font color="Maroon" title="c3">x</font> as    <a href="osafree.html#M2" title="OSAFREE:mode.2">MinTerm</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">X</font> ;<br/><a NAME="E3:95_1"/>
<span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">t</font>
 <b>by </b><i><a class="ref" href="osafree.html#D33" target="_self" title="OSAFREE:def.33">Def33</a></i>;<br/><b>hence </b><a NAME="E4:95_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osafree.html#K30" title="OSAFREE:func.30">MinTerms</a> <font color="Maroon" title="c2">X</font>
 <b>by </b><i><a class="ref" href="funct_2.html#T6" title="FUNCT_2:th.6">FUNCT_2:6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:95"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osafree.html#K30" title="OSAFREE:func.30">MinTerms</a> <font color="Maroon" title="c2">X</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> is    <a href="osafree.html#M2" title="OSAFREE:mode.2">MinTerm</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">X</font></span><br/>

<b>then consider </b><font color="Maroon" title="c4">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:95"/><i><font color="Green" title="E65">A1</font></i>: 
( <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">t</font> = <font color="Maroon" title="c4">y</font> as    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <b>by </b><i><a class="txt" href="#E4:95"><i><font color="Green" title="E65">A1</font></i></a></i>;<br/>
<a NAME="E6:95"/>
<span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">t</font> is    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">tx</font> = <font color="Maroon" title="c3">x</font> as    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <b>by </b><i><a class="txt" href="#E4:95"><i><font color="Green" title="E65">A1</font></i></a></i>;<br/>
<a NAME="E8:95"/>
<span class="p1">(<span class="default"><a href="osafree.html#K29" title="OSAFREE:func.29">PTMin</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">tx</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">tx</font>
 
<b>by </b><i><a class="txt" href="#E4:95"><i><font color="Green" title="E65">A1</font></i></a>, <a class="ref" href="osafree.html#T44" target="_self" title="OSAFREE:th.44">Th44</a></i>;<br/>
<b>hence </b><a NAME="E9:95"/>
<font color="Maroon" title="c3">x</font> is    <a href="osafree.html#M2" title="OSAFREE:mode.2">MinTerm</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">X</font>
 <b>by </b><i><a class="ref" href="osafree.html#D33" target="_self" title="OSAFREE:def.33">Def33</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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