<?xml version="1.0"?>
<div class="add">

<b>set </b><font color="Maroon" title="c3">FM</font> =  <a href="osafree.html#K16" title="OSAFREE:func.16">FreeOSA</a> <font color="Maroon" title="c2">X</font>;<br/>
<b>set </b><font color="Maroon" title="c4">D</font> =  <a href="osafree.html#K2" title="OSAFREE:func.2">DTConOSA</a> <font color="Maroon" title="c2">X</font>;<br/>
<b>set </b><font color="Maroon" title="c5">PTA</font> =  <a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font>;<br/>
<b>set </b><font color="Maroon" title="c6">SO</font> = the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K16" title="OSAFREE:func.16">FreeOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c7">NH</font> =  <a href="osalg_4.html#K18" title="OSALG_4:func.18">OSNat_Hom</a> <span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>,<span class="p1">(<span class="default"><a href="osafree.html#K15" title="OSAFREE:func.15">LCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">NH1</font> =  <a href="osalg_4.html#K18" title="OSALG_4:func.18">OSNat_Hom</a> <span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>,<span class="p1">(<span class="default"><a href="osafree.html#K15" title="OSAFREE:func.15">LCongruence</a> <font color="Maroon" title="c2">X</font></span>)</span> as   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>,<span class="p1">(<span class="default"><a href="osafree.html#K16" title="OSAFREE:func.16">FreeOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> ;<br/>
<a NAME="E2:69_1_1"/><i><font color="Green" title="E45">A1</font></i>: 
( <font color="Maroon" title="c8">NH1</font> <a href="msualg_3.html#R2" title="MSUALG_3:pred.2">is_epimorphism</a>  <a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font>, <a href="osafree.html#K16" title="OSAFREE:func.16">FreeOSA</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c8">NH1</font> is <a href="osalg_3.html#V1" title="OSALG_3:attr.1">order-sorted</a> )
 
<b>by </b><i><a class="ref" href="osalg_4.html#T16" title="OSALG_4:th.16">OSALG_4:16</a></i>;<br/>
<a NAME="E3:69_1_1"/><b>then </b><i><font color="Green" title="E46">A2</font></i>: 
( <font color="Maroon" title="c8">NH1</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a>  <a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font>, <a href="osafree.html#K16" title="OSAFREE:func.16">FreeOSA</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c8">NH1</font> is <a href="msualg_3.html#V2" title="MSUALG_3:attr.2">"onto"</a> )
 
<b>by </b><i><a class="ref" href="msualg_3.html#D10" title="MSUALG_3:def.10">MSUALG_3:def 10</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">SOS</font> = the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K16" title="OSAFREE:func.16">FreeOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> as   <a href="osalg_1.html#NM4" title="OSALG_1:NM.4">OrderSortedSet</a> of <font color="Maroon" title="c1">S</font> ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>) <b>-&gt; </b>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p2">(<span class="default"><a href="osafree.html#K16" title="OSAFREE:func.16">FreeOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span> =  <a href="osafree.html#K23" title="OSAFREE:func.23">OSFreeGen</a> $1,<font color="Maroon" title="c2">X</font>;<br/>
<b>consider </b><font color="Maroon" title="c10">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E6:69_1_1"/><i><font color="Green" title="E47">A3</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c10">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> &amp; ( for <font color="Olive" title="b1">s</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> holds  <font color="Maroon" title="c10">f</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">H<sub>1</sub></font>(<font color="Olive" title="b1">s</font>) ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S4" title="FUNCT_1:sch.4">FUNCT_1:sch 4</a>();<br/></i>
<b>reconsider </b><font color="Maroon" title="c11">f</font> = <font color="Maroon" title="c10">f</font> as    <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <b>by </b><i><a class="txt" href="#E6:69_1_1"><i><font color="Green" title="E47">A3</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E8:69_1_1"/><i><font color="Green" title="E48">A4</font></i>: 
<font color="Maroon" title="c11">f</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K16" title="OSAFREE:func.16">FreeOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c12">x</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">S</font> or <font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K16" title="OSAFREE:func.16">FreeOSA</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="c12">x</font> )</span><br/>

<b>assume </b><a NAME="E1:69_1_1_1"/>
<font color="Maroon" title="c12">x</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">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K16" title="OSAFREE:func.16">FreeOSA</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="c12">x</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c13">s</font> = <font color="Maroon" title="c12">x</font> as   <a href="osalg_1.html#NM1" title="OSALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S</font> ;<br/>
<a NAME="E3:69_1_1_1"/>
( <font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osafree.html#K23" title="OSAFREE:func.23">OSFreeGen</a> <font color="Maroon" title="c13">s</font>,<font color="Maroon" title="c2">X</font> &amp;  <a href="osafree.html#K23" title="OSAFREE:func.23">OSFreeGen</a> <font color="Maroon" title="c13">s</font>,<font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K16" title="OSAFREE:func.16">FreeOSA</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="c13">s</font> )
 
<b>by </b><i><a class="txt" href="#E6:69_1_1"><i><font color="Green" title="E47">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:69_1_1_1"/>
<font color="Maroon" title="c11">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K16" title="OSAFREE:func.16">FreeOSA</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="c12">x</font>
 ; <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>then reconsider </b><font color="Maroon" title="c12">f</font> = <font color="Maroon" title="c11">f</font> as   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <span class="p1">(<span class="default"><a href="osafree.html#K16" title="OSAFREE:func.16">FreeOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <b>by </b><i><a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E10:69_1_1"/>
 <a href="osalg_2.html#K2" title="OSALG_2:func.2">OSCl</a> <font color="Maroon" title="c12">f</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c9">SOS</font>
 
<b>by </b><i><a class="txt" href="#E8:69_1_1"><i><font color="Green" title="E48">A4</font></i></a>, <a class="ref" href="osalg_2.html#T13" title="OSALG_2:th.13">OSALG_2:13</a></i>;<br/>
<a NAME="E11:69_1_1"/><b>then </b>
 <a href="osalg_2.html#K2" title="OSALG_2:func.2">OSCl</a> <font color="Maroon" title="c12">f</font> is    <a href="pboole.html#M4" title="PBOOLE:mode.4">ManySortedSubset</a> of the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K16" title="OSAFREE:func.16">FreeOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c13">O</font> =  <a href="osalg_2.html#K2" title="OSALG_2:func.2">OSCl</a> <font color="Maroon" title="c12">f</font> as    <a href="osalg_2.html#M2" title="OSALG_2:mode.2">OSSubset</a> of  <a href="osafree.html#K16" title="OSAFREE:func.16">FreeOSA</a> <font color="Maroon" title="c2">X</font> <b>by </b><i><a class="ref" href="osalg_2.html#D2" title="OSALG_2:def.2">OSALG_2:def 2</a></i>;<br/>
<a NAME="E13:69_1_1"/>
<font color="Maroon" title="c13">O</font> is    <a href="osalg_2.html#M2" title="OSALG_2:mode.2">OSSubset</a> of  <a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Maroon" title="c13">O</font>
 
<b>by </b><i><a class="ref" href="osalg_2.html#D13" title="OSALG_2:def.13">OSALG_2:def 13</a></i>;<br/>
<a NAME="E14:69_1_1"/><b>then </b><i><font color="Green" title="E49">A5</font></i>: 
<font color="Maroon" title="c13">O</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Maroon" title="c13">O</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E15:69_1_1"/>
<font color="Maroon" title="c12">f</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c13">O</font>
 
<b>by </b><i><a class="ref" href="osalg_2.html#T12" title="OSALG_2:th.12">OSALG_2:12</a></i>;<br/>
<a NAME="E16:69_1_1"/><b>then </b><i><font color="Green" title="E50">A6</font></i>: 
<font color="Maroon" title="c12">f</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Maroon" title="c13">O</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E14:69_1_1"><i><font color="Green" title="E49">A5</font></i></a>, <a class="ref" href="pboole.html#T15" title="PBOOLE:th.15">PBOOLE:15</a></i>;<br/>
<a NAME="E17:69_1_1"/><i><font color="Green" title="E51">A7</font></i>: 
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Maroon" title="c13">O</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K16" title="OSAFREE:func.16">FreeOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:69_1_1_2"/>
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Maroon" title="c13">O</font></span>)</span> is   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <span class="p1">(<span class="default"><a href="osafree.html#K16" title="OSAFREE:func.16">FreeOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="msualg_2.html#D10" title="MSUALG_2:def.10">MSUALG_2:def 10</a></i>;<br/>
<b>hence </b><a NAME="E2:69_1_1_2"/>
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Maroon" title="c13">O</font></span>)</span> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K16" title="OSAFREE:func.16">FreeOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>
 <b>by </b><i><a class="ref" href="pboole.html#D23" title="PBOOLE:def.23">PBOOLE:def 23</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D13" title="PBOOLE:def.13">PBOOLE:def 13</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K16" title="OSAFREE:func.16">FreeOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Maroon" title="c13">O</font></span>)</span></span><br/>

<b>reconsider </b><font color="Maroon" title="c14">O2</font> = the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Maroon" title="c13">O</font></span>)</span> as   <a href="osalg_1.html#NM4" title="OSALG_1:NM.4">OrderSortedSet</a> of <font color="Maroon" title="c1">S</font> <b>by </b><i><a class="ref" href="osalg_1.html#T17" title="OSALG_1:th.17">OSALG_1:17</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>for <font color="Olive" title="b1">s1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>  st $1 <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"><font color="Maroon" title="c8">NH1</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Olive" title="b1">s1</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c8">NH1</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Olive" title="b1">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Maroon" title="c13">O</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s1</font>;<br/>
<a NAME="E4:69_1_1_2"/><i><font color="Green" title="E52">A8</font></i>: 
for <font color="Olive" title="b1">s</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <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>  st <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dtconstr.html#K6" title="DTCONSTR:func.6">Terminals</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> holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Olive" title="b1">s</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">t</font> be   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c15">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dtconstr.html#K6" title="DTCONSTR:func.6">Terminals</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> implies <font color="Maroon">S<sub>1</sub></font>[ <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c15">t</font>] )</span><br/>

<b>assume </b><a NAME="E1:69_1_1_2_1"/><i><font color="Green" title="E53">A9</font></i>: 
<font color="Maroon" title="c15">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dtconstr.html#K6" title="DTCONSTR:func.6">Terminals</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[ <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c15">t</font>]</span><br/>

<b>let </b><font color="Maroon" title="c16">s1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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"> (  <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c15">t</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"><font color="Maroon" title="c8">NH1</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c16">s1</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c8">NH1</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c16">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c15">t</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Maroon" title="c13">O</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">s1</font> )</span><br/>

<b>assume </b><a NAME="E2:69_1_1_2_1"/>
 <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c15">t</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"><font color="Maroon" title="c8">NH1</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c16">s1</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c8">NH1</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c16">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c15">t</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Maroon" title="c13">O</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">s1</font></span><br/>

<a NAME="E3:69_1_1_2_1"/><b>then </b>
 <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c15">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</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="c16">s1</font>
 
;<br/>
<a NAME="E4:69_1_1_2_1"/><b>then </b>
 <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c15">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osafree.html#K4" title="OSAFREE:func.4">ParsedTerms</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c16">s1</font>
 
<b>by </b><i><a class="ref" href="osafree.html#D8" target="_self" title="OSAFREE:def.8">Def8</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c17">a</font> being    <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> such that </b><br/><a NAME="E6:69_1_1_2_1"/><i><font color="Green" title="E54">A10</font></i>: 
 <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c15">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">a</font>
 <b>and </b><br/><a NAME="E7:69_1_1_2_1"/><i><font color="Green" title="E55">A11</font></i>: 
(  ex <font color="Olive" title="b1">s2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> ex <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">s2</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c16">s1</font> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s2</font> &amp; <font color="Maroon" title="c17">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">s2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) or  ex <font color="Olive" title="b1">o</font> being  <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Olive" title="b1">o</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c16">s1</font> ) )
 ;<br/>
<a NAME="E8:69_1_1_2_1"/>
for <font color="Olive" title="b1">o</font> being  <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font>  holds <br/>(  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  not  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Olive" title="b1">o</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c16">s1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon" title="c18">o</font> being   <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font><b> such that </b><a NAME="E2:69_1_1_2_1_1"/><i><font color="Green" title="E56">A12</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c18">o</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c16">s1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:69_1_1_2_1_1"/><i><font color="Green" title="E57">A13</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">t</font>
 
<b>by </b><i><a class="txt" href="#E6:69_1_1_2_1"><i><font color="Green" title="E54">A10</font></i></a>, <a class="txt" href="#E2:69_1_1_2_1_1"><i><font color="Green" title="E56">A12</font></i></a>, <a class="ref" href="trees_4.html#T3" title="TREES_4:th.3">TREES_4:3</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c19">s3</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c20">x3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:69_1_1_2_1_1"/><i><font color="Green" title="E58">A14</font></i>: 
( <font color="Maroon" title="c20">x3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">s3</font> &amp; <font color="Maroon" title="c15">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c20">x3</font>,<font color="Maroon" title="c19">s3</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="txt" href="#E1:69_1_1_2_1"><i><font color="Green" title="E53">A9</font></i></a>, <a class="ref" href="osafree.html#T4" target="_self" title="OSAFREE:th.4">Th4</a></i>;<br/>
<a NAME="E6:69_1_1_2_1_1"/>
( <font color="Maroon" title="c18">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">x3</font> &amp; the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">s3</font> )
 
<b>by </b><i><a class="txt" href="#E3:69_1_1_2_1_1"><i><font color="Green" title="E57">A13</font></i></a>, <a class="txt" href="#E5:69_1_1_2_1_1"><i><font color="Green" title="E58">A14</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E7:69_1_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c19">s3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c19">s3</font>
 
;<br/>
<b>hence </b><a NAME="E8:69_1_1_2_1_1"/>
contradiction
 ; <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>then consider </b><font color="Maroon" title="c18">s2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c19">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E10:69_1_1_2_1"/><i><font color="Green" title="E56">A15</font></i>: 
( <font color="Maroon" title="c18">s2</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c16">s1</font> &amp; <font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">s2</font> &amp; <font color="Maroon" title="c17">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">x</font>,<font color="Maroon" title="c18">s2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="txt" href="#E7:69_1_1_2_1"><i><font color="Green" title="E55">A11</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c20">s11</font> = <font color="Maroon" title="c16">s1</font>, <font color="Maroon" title="c21">s21</font> = <font color="Maroon" title="c18">s2</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> ;<br/>
<a NAME="E12:69_1_1_2_1"/><i><font color="Green" title="E57">A16</font></i>: 
<font color="Maroon" title="c14">O2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">s21</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">O2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">s11</font>
 
<b>by </b><i><a class="txt" href="#E10:69_1_1_2_1"><i><font color="Green" title="E56">A15</font></i></a>, <a class="ref" href="osalg_1.html#D18" title="OSALG_1:def.18">OSALG_1:def 18</a></i>;<br/>
<a NAME="E13:69_1_1_2_1"/><i><font color="Green" title="E58">A17</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">NH1</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c18">s2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">x</font>,<font color="Maroon" title="c18">s2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osafree.html#K23" title="OSAFREE:func.23">OSFreeGen</a> <font color="Maroon" title="c18">s2</font>,<font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="txt" href="#E10:69_1_1_2_1"><i><font color="Green" title="E56">A15</font></i></a>, <a class="ref" href="osafree.html#D26" target="_self" title="OSAFREE:def.26">Def26</a></i>;<br/>
<a NAME="E14:69_1_1_2_1"/>
<font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osafree.html#K4" title="OSAFREE:func.4">ParsedTerms</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">s2</font>
 
<b>by </b><i><a class="txt" href="#E10:69_1_1_2_1"><i><font color="Green" title="E56">A15</font></i></a></i>;<br/>
<a NAME="E15:69_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c17">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="osafree.html#K5" title="OSAFREE:func.5">ParsedTerms</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="c18">s2</font>
 
<b>by </b><i><a class="ref" href="osafree.html#D8" target="_self" title="OSAFREE:def.8">Def8</a></i>;<br/>
<a NAME="E16:69_1_1_2_1"/><b>then </b>
 <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">x</font>,<font color="Maroon" title="c18">s2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <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"><font color="Maroon" title="c8">NH1</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c18">s2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E10:69_1_1_2_1"><i><font color="Green" title="E56">A15</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E17:69_1_1_2_1"/><b>then </b><i><font color="Green" title="E59">A18</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">NH1</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c21">s21</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">x</font>,<font color="Maroon" title="c18">s2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">NH1</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c20">s11</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">x</font>,<font color="Maroon" title="c18">s2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E2:69_1_1"><i><font color="Green" title="E45">A1</font></i></a>, <a class="txt" href="#E10:69_1_1_2_1"><i><font color="Green" title="E56">A15</font></i></a>, <a class="ref" href="osalg_3.html#D1" title="OSALG_3:def.1">OSALG_3:def 1</a></i>;<br/>
<a NAME="E18:69_1_1_2_1"/>
<font color="Maroon" title="c12">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">s2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Maroon" title="c13">O</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">s2</font>
 
<b>by </b><i><a class="txt" href="#E16:69_1_1"><i><font color="Green" title="E50">A6</font></i></a>, <a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a></i>;<br/>
<a NAME="E19:69_1_1_2_1"/><b>then </b>
 <a href="osafree.html#K23" title="OSAFREE:func.23">OSFreeGen</a> <font color="Maroon" title="c18">s2</font>,<font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Maroon" title="c13">O</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">s2</font>
 
<b>by </b><i><a class="txt" href="#E6:69_1_1"><i><font color="Green" title="E47">A3</font></i></a></i>;<br/>
<a NAME="E20:69_1_1_2_1"/><b>then </b>
 <a href="osafree.html#K23" title="OSAFREE:func.23">OSFreeGen</a> <font color="Maroon" title="c18">s2</font>,<font color="Maroon" title="c2">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Maroon" title="c13">O</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">s1</font>
 
<b>by </b><i><a class="txt" href="#E12:69_1_1_2_1"><i><font color="Green" title="E57">A16</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E21:69_1_1_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">NH1</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c16">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c15">t</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Maroon" title="c13">O</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">s1</font>
 <b>by </b><i><a class="txt" href="#E6:69_1_1_2_1"><i><font color="Green" title="E54">A10</font></i></a>, <a class="txt" href="#E10:69_1_1_2_1"><i><font color="Green" title="E56">A15</font></i></a>, <a class="txt" href="#E13:69_1_1_2_1"><i><font color="Green" title="E58">A17</font></i></a>, <a class="txt" href="#E17:69_1_1_2_1"><i><font color="Green" title="E59">A18</font></i></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>
<a NAME="E5:69_1_1_2"/><i><font color="Green" title="E53">A19</font></i>: 
for <font color="Olive" title="b1">nt</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <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/> for <font color="Olive" title="b2">ts</font> being   <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</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>  st <font color="Olive" title="b1">nt</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Olive" title="b2">ts</font> &amp; ( for <font color="Olive" title="b3">t</font> being  <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <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>  st <font color="Olive" title="b3">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b2">ts</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b3">t</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Olive" title="b2">ts</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c15">nt</font> be   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <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>; <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">ts</font> being   <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</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>  st <font color="Maroon" title="c15">nt</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Olive" title="b1">ts</font> &amp; ( for <font color="Olive" title="b2">t</font> being  <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <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>  st <font color="Olive" title="b2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">ts</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">t</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c15">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Olive" title="b1">ts</font>]</span><br/><b>let </b><font color="Maroon" title="c16">ts</font> be    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c15">nt</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c16">ts</font> &amp; ( for <font color="Olive" title="b1">t</font> being  <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <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>  st <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c16">ts</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">t</font>] ) implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c15">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c16">ts</font>] )</span><br/>



<b>assume </b><a NAME="E1:69_1_1_2_2"/><i><font color="Green" title="E54">A20</font></i>: 
( <font color="Maroon" title="c15">nt</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c16">ts</font> &amp; ( for <font color="Olive" title="b1">t</font> being  <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <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>  st <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c16">ts</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">t</font>] ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c15">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c16">ts</font>]</span><br/>

<b>set </b><font color="Maroon" title="c17">G</font> =  <a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Maroon" title="c13">O</font>;<br/>
<b>set </b><font color="Maroon" title="c18">OU</font> = <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">S</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p2">(<span class="default"><a href="msafree.html#K3" title="MSAFREE:func.3">coprod</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span>;<br/>
<a NAME="E2:69_1_1_2_2"/><i><font color="Green" title="E55">A21</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">nt</font>,<span class="p2">(<span class="default"><a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c16">ts</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="lang1.html#U1" title="LANG1:sel.1">Rules</a> of <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="#E1:69_1_1_2_2"><i><font color="Green" title="E54">A20</font></i></a>, <a class="ref" href="lang1.html#D1" title="LANG1:def.1">LANG1:def 1</a></i>;<br/>
<a NAME="E3:69_1_1_2_2"/><i><font color="Green" title="E56">A22</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">nt</font>,<span class="p2">(<span class="default"><a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c16">ts</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osafree.html#K1" title="OSAFREE:func.1">OSREL</a> <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="txt" href="#E1:69_1_1_2_2"><i><font color="Green" title="E54">A20</font></i></a>, <a class="ref" href="lang1.html#D1" title="LANG1:def.1">LANG1:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c19">sy</font> = <font color="Maroon" title="c15">nt</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">S</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p2">(<span class="default"><a href="msafree.html#K3" title="MSAFREE:func.3">coprod</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c20">rt</font> =  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c16">ts</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">S</font>,<span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p3">(<span class="default"><a href="msafree.html#K3" title="MSAFREE:func.3">coprod</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <b>by </b><i><a class="txt" href="#E2:69_1_1_2_2"><i><font color="Green" title="E55">A21</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E6:69_1_1_2_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">sy</font>,<font color="Maroon" title="c20">rt</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osafree.html#K1" title="OSAFREE:func.1">OSREL</a> <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="txt" href="#E1:69_1_1_2_2"><i><font color="Green" title="E54">A20</font></i></a>, <a class="ref" href="lang1.html#D1" title="LANG1:def.1">LANG1:def 1</a></i>;<br/>
<a NAME="E7:69_1_1_2_2"/><b>then </b>
<font color="Maroon" title="c19">sy</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">S</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="osafree.html#D4" target="_self" title="OSAFREE:def.4">Def4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c21">o</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c22">x2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span><b> such that </b><br/><a NAME="E9:69_1_1_2_2"/><i><font color="Green" title="E57">A23</font></i>: 
<font color="Maroon" title="c19">sy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">o</font>,<font color="Maroon" title="c22">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="domain_1.html#T9" title="DOMAIN_1:th.9">DOMAIN_1:9</a></i>;<br/>
<a NAME="E10:69_1_1_2_2"/><i><font color="Green" title="E58">A24</font></i>: 
<font color="Maroon" title="c22">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>set </b><font color="Maroon" title="c23">ar</font> =  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c21">o</font>;<br/>
<b>set </b><font color="Maroon" title="c24">rs</font> =  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c21">o</font>;<br/>
<b>let </b><font color="Maroon" title="c25">s1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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"> ( <font color="Maroon" title="c15">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c16">ts</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"><font color="Maroon" title="c8">NH1</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c25">s1</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c8">NH1</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c25">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c16">ts</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Maroon" title="c13">O</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">s1</font> )</span><br/>

<b>assume </b><a NAME="E11:69_1_1_2_2"/><i><font color="Green" title="E59">A25</font></i>: 
<font color="Maroon" title="c15">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c16">ts</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"><font color="Maroon" title="c8">NH1</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c25">s1</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c8">NH1</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c25">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c16">ts</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Maroon" title="c13">O</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">s1</font></span><br/>

<a NAME="E12:69_1_1_2_2"/>
<font color="Maroon" title="c15">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c16">ts</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</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="c25">s1</font>
 
<b>by </b><i><a class="txt" href="#E11:69_1_1_2_2"><i><font color="Green" title="E59">A25</font></i></a></i>;<br/>
<a NAME="E13:69_1_1_2_2"/><b>then </b>
<font color="Maroon" title="c15">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c16">ts</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="osafree.html#K4" title="OSAFREE:func.4">ParsedTerms</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c25">s1</font>
 
<b>by </b><i><a class="ref" href="osafree.html#D8" target="_self" title="OSAFREE:def.8">Def8</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c26">a1</font> being    <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> such that </b><br/><a NAME="E15:69_1_1_2_2"/><i><font color="Green" title="E60">A26</font></i>: 
<font color="Maroon" title="c15">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c16">ts</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">a1</font>
 <b>and </b><br/><a NAME="E16:69_1_1_2_2"/><i><font color="Green" title="E61">A27</font></i>: 
(  ex <font color="Olive" title="b1">s2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> ex <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">s2</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c25">s1</font> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s2</font> &amp; <font color="Maroon" title="c26">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">s2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) or  ex <font color="Olive" title="b1">o</font> being  <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Olive" title="b1">o</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c25">s1</font> ) )
 ;<br/>
<a NAME="E17:69_1_1_2_2"/>
for <font color="Olive" title="b1">s2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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/>(  not <font color="Olive" title="b1">s2</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c25">s1</font> or  not <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s2</font> or  not <font color="Maroon" title="c26">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">s2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_2_2_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon" title="c27">s2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c28">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><a NAME="E2:69_1_1_2_2_1"/><i><font color="Green" title="E62">A28</font></i>: 
( <font color="Maroon" title="c27">s2</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c25">s1</font> &amp; <font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">s2</font> &amp; <font color="Maroon" title="c26">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c28">x</font>,<font color="Maroon" title="c27">s2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c28">x</font>,<font color="Maroon" title="c27">s2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c16">ts</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
<b>by </b><i><a class="txt" href="#E15:69_1_1_2_2"><i><font color="Green" title="E60">A26</font></i></a>, <a class="txt" href="#E2:69_1_1_2_2_1"><i><font color="Green" title="E62">A28</font></i></a>, <a class="ref" href="trees_4.html#T3" title="TREES_4:th.3">TREES_4:3</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">o</font>,<font color="Maroon" title="c22">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="#E9:69_1_1_2_2"><i><font color="Green" title="E57">A23</font></i></a>, <a class="ref" href="trees_4.html#D4" title="TREES_4:def.4">TREES_4:def 4</a></i>
;<br/>
<a NAME="E4:69_1_1_2_2_1"/><b>then </b>
<font color="Maroon" title="c27">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>
 
<b>by </b><i><a class="txt" href="#E10:69_1_1_2_2"><i><font color="Green" title="E58">A24</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E5:69_1_1_2_2_1"/><b>then </b>
<font color="Maroon" title="c27">s2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c27">s2</font>
 
;<br/>
<b>hence </b><a NAME="E6:69_1_1_2_2_1"/>
contradiction
 ; <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>then consider </b><font color="Maroon" title="c27">o1</font> being   <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font><b> such that </b><br/><a NAME="E19:69_1_1_2_2"/><i><font color="Green" title="E62">A29</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c27">o1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">a1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c27">o1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c25">s1</font> )
 <b>by </b><i><a class="txt" href="#E16:69_1_1_2_2"><i><font color="Green" title="E61">A27</font></i></a></i>;<br/>
<a NAME="E20:69_1_1_2_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c27">o1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">o</font>,<font color="Maroon" title="c22">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="#E9:69_1_1_2_2"><i><font color="Green" title="E57">A23</font></i></a>, <a class="txt" href="#E15:69_1_1_2_2"><i><font color="Green" title="E60">A26</font></i></a>, <a class="txt" href="#E19:69_1_1_2_2"><i><font color="Green" title="E62">A29</font></i></a>, <a class="ref" href="trees_4.html#D4" title="TREES_4:def.4">TREES_4:def 4</a></i>;<br/>
<a NAME="E21:69_1_1_2_2"/><b>then </b><i><font color="Green" title="E63">A30</font></i>: 
<font color="Maroon" title="c27">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">o</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c28">B</font> = the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Maroon" title="c13">O</font></span>)</span> as   <a href="msualg_2.html#NM1" title="MSUALG_2:NM.1">MSSubset</a> of <span class="p1">(<span class="default"><a href="osafree.html#K16" title="OSAFREE:func.16">FreeOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <b>by </b><i><a class="ref" href="msualg_2.html#D10" title="MSUALG_2:def.10">MSUALG_2:def 10</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c29">B1</font> = <font color="Maroon" title="c28">B</font> as   <a href="osalg_1.html#NM4" title="OSALG_1:NM.4">OrderSortedSet</a> of <font color="Maroon" title="c1">S</font> <b>by </b><i><a class="ref" href="osalg_1.html#T17" title="OSALG_1:th.17">OSALG_1:17</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c30">rs1</font> =  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c21">o</font>, <font color="Maroon" title="c31">s11</font> = <font color="Maroon" title="c25">s1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> ;<br/>
<a NAME="E25:69_1_1_2_2"/><i><font color="Green" title="E64">A31</font></i>: 
<font color="Maroon" title="c29">B1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">rs1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c29">B1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">s11</font>
 
<b>by </b><i><a class="txt" href="#E19:69_1_1_2_2"><i><font color="Green" title="E62">A29</font></i></a>, <a class="txt" href="#E21:69_1_1_2_2"><i><font color="Green" title="E63">A30</font></i></a>, <a class="ref" href="osalg_1.html#D18" title="OSALG_1:def.18">OSALG_1:def 18</a></i>;<br/>
<b>set </b><font color="Maroon" title="c32">AR</font> = the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c1">S</font>;<br/>
<b>set </b><font color="Maroon" title="c33">RS</font> = the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c1">S</font>;<br/>
<b>set </b><font color="Maroon" title="c34">BH</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c28">B</font> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c1">S</font>;<br/>
<b>set </b><font color="Maroon" title="c35">Op</font> = the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">S</font>;<br/>
<a NAME="E26:69_1_1_2_2"/>
<font color="Maroon" title="c28">B</font> is <a href="msualg_2.html#V3" title="MSUALG_2:attr.3">opers_closed</a>
 
<b>by </b><i><a class="ref" href="msualg_2.html#D10" title="MSUALG_2:def.10">MSUALG_2:def 10</a></i>;<br/>
<a NAME="E27:69_1_1_2_2"/><b>then </b>
<font color="Maroon" title="c28">B</font> <a href="msualg_2.html#R1" title="MSUALG_2:pred.1">is_closed_on</a> <font color="Maroon" title="c21">o</font>
 
<b>by </b><i><a class="ref" href="msualg_2.html#D7" title="MSUALG_2:def.7">MSUALG_2:def 7</a></i>;<br/>
<a NAME="E28:69_1_1_2_2"/><b>then </b><i><font color="Green" title="E65">A32</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c21">o</font>,<span class="p3">(<span class="default"><a href="osafree.html#K16" title="OSAFREE:func.16">FreeOSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c28">B</font> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">o</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">B</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">o</font>
 
<b>by </b><i><a class="ref" href="msualg_2.html#D6" title="MSUALG_2:def.6">MSUALG_2:def 6</a></i>;<br/>
<a NAME="E29:69_1_1_2_2"/><i><font color="Green" title="E66">A33</font></i>: 
(  <a href="osafree.html#K9" title="OSAFREE:func.9">OSSym</a> <font color="Maroon" title="c21">o</font>,<font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c15">nt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 
<b>by </b><i><a class="txt" href="#E9:69_1_1_2_2"><i><font color="Green" title="E57">A23</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<i><font color="Green" title="E67">A34</font></i>:  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c21">o</font>,<span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="osafree.html#K7" title="OSAFREE:func.7">PTOper</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="msualg_4.html#K5" title="MSUALG_4:func.5">.</a> <font color="Maroon" title="c21">o</font>
<b>by </b><i><a class="ref" href="msualg_1.html#D11" title="MSUALG_1:def.11">MSUALG_1:def 11</a></i>
<br/>.= 
 <a href="osafree.html#K6" title="OSAFREE:func.6">PTDenOp</a> <font color="Maroon" title="c21">o</font>,<font color="Maroon" title="c2">X</font>
<b>by </b><i><a class="ref" href="osafree.html#D10" target="_self" title="OSAFREE:def.10">Def10</a></i>
;<br/>
<a NAME="E31:69_1_1_2_2"/><i><font color="Green" title="E68">A35</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="osafree.html#K5" title="OSAFREE:func.5">ParsedTerms</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> &amp; <font color="Maroon" title="c21">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">S</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c28">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">S</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c1">S</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="c1">S</font> &amp; the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c1">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c21">o</font> &amp; the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c1">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c21">o</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="msualg_1.html#D6" title="MSUALG_1:def.6">MSUALG_1:def 6</a>, <a class="ref" href="msualg_1.html#D7" title="MSUALG_1:def.7">MSUALG_1:def 7</a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a>, <a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/>
<a NAME="E32:69_1_1_2_2"/><i><font color="Green" title="E69">A36</font></i>: 
<font color="Maroon" title="c16">ts</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="osafree.html#K5" title="OSAFREE:func.5">ParsedTerms</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">o</font>
 
<b>by </b><i><a class="txt" href="#E1:69_1_1_2_2"><i><font color="Green" title="E54">A20</font></i></a>, <a class="txt" href="#E29:69_1_1_2_2"><i><font color="Green" title="E66">A33</font></i></a>, <a class="ref" href="osafree.html#T7" target="_self" title="OSAFREE:th.7">Th7</a></i>;<br/>
<a NAME="E33:69_1_1_2_2"/><b>then </b><i><font color="Green" title="E70">A37</font></i>: 
<font color="Maroon" title="c16">ts</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c21">o</font>,<span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="msualg_1.html#D9" title="MSUALG_1:def.9">MSUALG_1:def 9</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c36">ts1</font> = <font color="Maroon" title="c16">ts</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c21">o</font>,<span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <b>by </b><i><a class="txt" href="#E32:69_1_1_2_2"><i><font color="Green" title="E69">A36</font></i></a>, <a class="ref" href="msualg_1.html#D9" title="MSUALG_1:def.9">MSUALG_1:def 9</a></i>;<br/>
<a NAME="E35:69_1_1_2_2"/><i><font color="Green" title="E71">A38</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">NH1</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c21">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c21">o</font>,<span class="p3">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c36">ts1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c21">o</font>,<span class="p2">(<span class="default"><a href="osafree.html#K16" title="OSAFREE:func.16">FreeOSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">NH1</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c36">ts1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E3:69_1_1"><i><font color="Green" title="E46">A2</font></i></a>, <a class="ref" href="msualg_3.html#D9" title="MSUALG_3:def.9">MSUALG_3:def 9</a></i>;<br/>
<a NAME="E36:69_1_1_2_2"/><i><font color="Green" title="E72">A39</font></i>: 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c21">o</font>,<span class="p2">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">ts</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c16">ts</font>
 
<b>by </b><i><a class="txt" href="#E1:69_1_1_2_2"><i><font color="Green" title="E54">A20</font></i></a>, <a class="txt" href="#E29:69_1_1_2_2"><i><font color="Green" title="E66">A33</font></i></a>, <a class="txt" href="#E30:69_1_1_2_2"><i><font color="Green" title="E67">A34</font></i></a>, <a class="ref" href="osafree.html#D9" target="_self" title="OSAFREE:def.9">Def9</a></i>;<br/>
<a NAME="E37:69_1_1_2_2"/>
<font color="Maroon" title="c8">NH1</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c36">ts1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c21">o</font>,<span class="p1">(<span class="default"><a href="osafree.html#K16" title="OSAFREE:func.16">FreeOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>
 
;<br/>
<a NAME="E38:69_1_1_2_2"/><b>then </b><i><font color="Green" title="E73">A40</font></i>: 
<font color="Maroon" title="c8">NH1</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c36">ts1</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="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c21">o</font>,<span class="p2">(<span class="default"><a href="osafree.html#K16" title="OSAFREE:func.16">FreeOSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E39:69_1_1_2_2"/><i><font color="Green" title="E74">A41</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">B</font> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">B</font> <a href="msualg_4.html#K6" title="MSUALG_4:func.6">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c21">o</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E31:69_1_1_2_2"><i><font color="Green" title="E68">A35</font></i></a>, <a class="ref" href="msafree.html#T1" title="MSAFREE:th.1">MSAFREE:1</a></i>;<br/>
<a NAME="E40:69_1_1_2_2"/><i><font color="Green" title="E75">A42</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">B</font> <a href="msualg_4.html#K6" title="MSUALG_4:func.6">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c21">o</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c21">o</font></span>)</span> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p2">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="msualg_4.html#K6" title="MSUALG_4:func.6">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c21">o</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c21">o</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E41:69_1_1_2_2"/><i><font color="Green" title="E76">A43</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c16">ts</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c16">ts</font>
 
<b>by </b><i><a class="ref" href="trees_3.html#D18" title="TREES_3:def.18">TREES_3:def 18</a></i>;<br/>
<a NAME="E42:69_1_1_2_2"/><i><font color="Green" title="E77">A44</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">rt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c21">o</font></span>)</span> &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">rt</font> holds <br/>( ( <font color="Maroon" title="c20">rt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">S</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> implies for <font color="Olive" title="b2">o1</font> being  <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">o1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">rt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> holds <br/> <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Olive" title="b2">o1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c21">o</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font> ) &amp; ( <font color="Maroon" title="c20">rt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="msafree.html#K3" title="MSAFREE:func.3">coprod</a> <font color="Maroon" title="c2">X</font></span>)</span> implies  ex <font color="Olive" title="b2">i</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> st <br/>( <font color="Olive" title="b2">i</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c21">o</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">x</font> &amp; <font color="Maroon" title="c20">rt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree.html#K2" title="MSAFREE:func.2">coprod</a> <font color="Olive" title="b2">i</font>,<font color="Maroon" title="c2">X</font> ) ) ) ) )
 
<b>by </b><i><a class="txt" href="#E3:69_1_1_2_2"><i><font color="Green" title="E56">A22</font></i></a>, <a class="txt" href="#E9:69_1_1_2_2"><i><font color="Green" title="E57">A23</font></i></a>, <a class="txt" href="#E10:69_1_1_2_2"><i><font color="Green" title="E58">A24</font></i></a>, <a class="ref" href="osafree.html#T2" target="_self" title="OSAFREE:th.2">Th2</a></i>;<br/>
<a NAME="E43:69_1_1_2_2"/><i><font color="Green" title="E78">A45</font></i>: 
(  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">rt</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c20">rt</font> &amp;  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c21">o</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c21">o</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E44:69_1_1_2_2"/><i><font color="Green" title="E79">A46</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">NH1</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c36">ts1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c21">o</font></span>)</span>
 
<b>by </b><i><a class="ref" href="msualg_3.html#T6" title="MSUALG_3:th.6">MSUALG_3:6</a></i>;<br/>
<a NAME="E45:69_1_1_2_2"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">B</font> <a href="msualg_4.html#K6" title="MSUALG_4:func.6">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c21">o</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c8">NH1</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c36">ts1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">B</font> <a href="msualg_4.html#K6" title="MSUALG_4:func.6">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c21">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_2_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c37">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="c37">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">B</font> <a href="msualg_4.html#K6" title="MSUALG_4:func.6">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c21">o</font></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c8">NH1</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c36">ts1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">B</font> <a href="msualg_4.html#K6" title="MSUALG_4:func.6">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c21">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">x</font> )</span><br/>

<b>assume </b><a NAME="E1:69_1_1_2_2_3"/><i><font color="Green" title="E80">A47</font></i>: 
<font color="Maroon" title="c37">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">B</font> <a href="msualg_4.html#K6" title="MSUALG_4:func.6">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c21">o</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c8">NH1</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c36">ts1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">B</font> <a href="msualg_4.html#K6" title="MSUALG_4:func.6">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c21">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">x</font></span><br/>

<b>reconsider </b><font color="Maroon" title="c38">x1</font> = <font color="Maroon" title="c37">x</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <b>by </b><i><a class="txt" href="#E1:69_1_1_2_2_3"><i><font color="Green" title="E80">A47</font></i></a></i>;<br/>
<a NAME="E3:69_1_1_2_2_3"/><i><font color="Green" title="E81">A48</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c16">ts</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <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="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E4:69_1_1_2_2_3"/><i><font color="Green" title="E82">A49</font></i>: 
<font color="Maroon" title="c16">ts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c38">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c16">ts</font>
 
<b>by </b><i><a class="txt" href="#E40:69_1_1_2_2"><i><font color="Green" title="E75">A42</font></i></a>, <a class="txt" href="#E41:69_1_1_2_2"><i><font color="Green" title="E76">A43</font></i></a>, <a class="txt" href="#E42:69_1_1_2_2"><i><font color="Green" title="E77">A44</font></i></a>, <a class="txt" href="#E43:69_1_1_2_2"><i><font color="Green" title="E78">A45</font></i></a>, <a class="txt" href="#E1:69_1_1_2_2_3"><i><font color="Green" title="E80">A47</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c39">t</font> = <font color="Maroon" title="c16">ts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">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="#E3:69_1_1_2_2_3"><i><font color="Green" title="E81">A48</font></i></a></i>;<br/>
<a NAME="E6:69_1_1_2_2_3"/><i><font color="Green" title="E83">A50</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">NH1</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c36">ts1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c38">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">NH1</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c21">o</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c38">x1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">ts1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c38">x1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E40:69_1_1_2_2"><i><font color="Green" title="E75">A42</font></i></a>, <a class="txt" href="#E41:69_1_1_2_2"><i><font color="Green" title="E76">A43</font></i></a>, <a class="txt" href="#E42:69_1_1_2_2"><i><font color="Green" title="E77">A44</font></i></a>, <a class="txt" href="#E43:69_1_1_2_2"><i><font color="Green" title="E78">A45</font></i></a>, <a class="txt" href="#E1:69_1_1_2_2_3"><i><font color="Green" title="E80">A47</font></i></a>, <a class="ref" href="msualg_3.html#D8" title="MSUALG_3:def.8">MSUALG_3:def 8</a></i>;<br/>
<a NAME="E7:69_1_1_2_2_3"/>
<font color="Maroon" title="c36">ts1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p2">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="msualg_4.html#K6" title="MSUALG_4:func.6">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c21">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">x</font>
 
<b>by </b><i><a class="txt" href="#E40:69_1_1_2_2"><i><font color="Green" title="E75">A42</font></i></a>, <a class="txt" href="#E1:69_1_1_2_2_3"><i><font color="Green" title="E80">A47</font></i></a>, <a class="ref" href="msualg_3.html#T6" title="MSUALG_3:th.6">MSUALG_3:6</a></i>;<br/>
<a NAME="E8:69_1_1_2_2_3"/><b>then </b>
<font color="Maroon" title="c36">ts1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c21">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E40:69_1_1_2_2"><i><font color="Green" title="E75">A42</font></i></a>, <a class="txt" href="#E1:69_1_1_2_2_3"><i><font color="Green" title="E80">A47</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/>
<a NAME="E9:69_1_1_2_2_3"/><b>then </b>
<font color="Maroon" title="c36">ts1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c21">o</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c37">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E40:69_1_1_2_2"><i><font color="Green" title="E75">A42</font></i></a>, <a class="txt" href="#E1:69_1_1_2_2_3"><i><font color="Green" title="E80">A47</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<a NAME="E10:69_1_1_2_2_3"/><b>then </b>
<font color="Maroon" title="c36">ts1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c38">x1</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"><font color="Maroon" title="c8">NH1</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c21">o</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c38">x1</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E11:69_1_1_2_2_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">NH1</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c21">o</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c38">x1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c39">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Maroon" title="c13">O</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c21">o</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c38">x1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:69_1_1_2_2"><i><font color="Green" title="E54">A20</font></i></a>, <a class="txt" href="#E4:69_1_1_2_2_3"><i><font color="Green" title="E82">A49</font></i></a></i>;<br/>
<a NAME="E12:69_1_1_2_2_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">NH1</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c21">o</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c38">x1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c36">ts1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c38">x1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c28">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c21">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">x</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E40:69_1_1_2_2"><i><font color="Green" title="E75">A42</font></i></a>, <a class="txt" href="#E1:69_1_1_2_2_3"><i><font color="Green" title="E80">A47</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>
<b>hence </b><a NAME="E13:69_1_1_2_2_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">NH1</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c36">ts1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">B</font> <a href="msualg_4.html#K6" title="MSUALG_4:func.6">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c21">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c37">x</font>
 <b>by </b><i><a class="txt" href="#E1:69_1_1_2_2_3"><i><font color="Green" title="E80">A47</font></i></a>, <a class="txt" href="#E6:69_1_1_2_2_3"><i><font color="Green" title="E83">A50</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</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>
<a NAME="E46:69_1_1_2_2"/><b>then </b>
<font color="Maroon" title="c8">NH1</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c36">ts1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c28">B</font> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">o</font>
 
<b>by </b><i><a class="txt" href="#E39:69_1_1_2_2"><i><font color="Green" title="E74">A41</font></i></a>, <a class="txt" href="#E40:69_1_1_2_2"><i><font color="Green" title="E75">A42</font></i></a>, <a class="txt" href="#E44:69_1_1_2_2"><i><font color="Green" title="E79">A46</font></i></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></i>;<br/>
<a NAME="E47:69_1_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c21">o</font>,<span class="p2">(<span class="default"><a href="osafree.html#K16" title="OSAFREE:func.16">FreeOSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">NH1</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c36">ts1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c21">o</font>,<span class="p3">(<span class="default"><a href="osafree.html#K16" title="OSAFREE:func.16">FreeOSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c28">B</font> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">o</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E38:69_1_1_2_2"><i><font color="Green" title="E73">A40</font></i></a>, <a class="ref" href="funct_1.html#T73" title="FUNCT_1:th.73">FUNCT_1:73</a></i>;<br/>
<a NAME="E48:69_1_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c21">o</font>,<span class="p2">(<span class="default"><a href="osafree.html#K16" title="OSAFREE:func.16">FreeOSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">NH1</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c36">ts1</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c28">B</font> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">o</font>
 
<b>by </b><i><a class="txt" href="#E28:69_1_1_2_2"><i><font color="Green" title="E65">A32</font></i></a></i>;<br/>
<a NAME="E49:69_1_1_2_2"/><b>then </b><i><font color="Green" title="E80">A51</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">NH1</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c21">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c16">ts</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c28">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c21">o</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E31:69_1_1_2_2"><i><font color="Green" title="E68">A35</font></i></a>, <a class="txt" href="#E35:69_1_1_2_2"><i><font color="Green" title="E71">A38</font></i></a>, <a class="txt" href="#E36:69_1_1_2_2"><i><font color="Green" title="E72">A39</font></i></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>;<br/>
<a NAME="E50:69_1_1_2_2"/>
<span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c21">o</font>,<span class="p2">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">ts</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_1.html#K4" title="MSUALG_1:func.4">Result</a> <font color="Maroon" title="c21">o</font>,<span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E33:69_1_1_2_2"><i><font color="Green" title="E70">A37</font></i></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/>
<a NAME="E51:69_1_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c21">o</font>,<span class="p2">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">ts</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p2">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="pboole.html#K7" title="PBOOLE:func.7">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">o</font>
 
<b>by </b><i><a class="ref" href="msualg_1.html#D10" title="MSUALG_1:def.10">MSUALG_1:def 10</a></i>;<br/>
<a NAME="E52:69_1_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c21">o</font>,<span class="p2">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">ts</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c21">o</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E31:69_1_1_2_2"><i><font color="Green" title="E68">A35</font></i></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>;<br/>
<a NAME="E53:69_1_1_2_2"/><b>then </b>
<font color="Maroon" title="c15">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c16">ts</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"><font color="Maroon" title="c8">NH1</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Maroon" title="c21">o</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E36:69_1_1_2_2"><i><font color="Green" title="E72">A39</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E54:69_1_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">NH1</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c31">s11</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c16">ts</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">NH1</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c30">rs1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c16">ts</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E2:69_1_1"><i><font color="Green" title="E45">A1</font></i></a>, <a class="txt" href="#E19:69_1_1_2_2"><i><font color="Green" title="E62">A29</font></i></a>, <a class="txt" href="#E21:69_1_1_2_2"><i><font color="Green" title="E63">A30</font></i></a>, <a class="ref" href="osalg_3.html#D1" title="OSALG_3:def.1">OSALG_3:def 1</a></i>;<br/>
<b>hence </b><a NAME="E55:69_1_1_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">NH1</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c25">s1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c16">ts</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Maroon" title="c13">O</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">s1</font>
 <b>by </b><i><a class="txt" href="#E25:69_1_1_2_2"><i><font color="Green" title="E64">A31</font></i></a>, <a class="txt" href="#E49:69_1_1_2_2"><i><font color="Green" title="E80">A51</font></i></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>
<a NAME="E6:69_1_1_2"/><i><font color="Green" title="E54">A52</font></i>: 
for <font color="Olive" title="b1">t</font> being  <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <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>  st <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <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> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">t</font>]
 
<b>from </b><i><a class="ref" href="dtconstr.html#S7" title="DTCONSTR:sch.7">DTCONSTR:sch 7</a>(<a class="txt" href="#E4:69_1_1_2"><i><font color="Green" title="E52">A8</font></i></a>, <a class="txt" href="#E5:69_1_1_2"><i><font color="Green" title="E53">A19</font></i></a>);<br/></i>
<b>let </b><font color="Maroon" title="c15">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D5" title="PBOOLE:def.5">PBOOLE:def 5</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c15">x</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">S</font> or the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K16" title="OSAFREE:func.16">FreeOSA</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="c15">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Maroon" title="c13">O</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font> )</span><br/>

<b>assume </b><a NAME="E7:69_1_1_2"/><i><font color="Green" title="E55">A53</font></i>: 
<font color="Maroon" title="c15">x</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">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K16" title="OSAFREE:func.16">FreeOSA</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="c15">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Maroon" title="c13">O</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c16">s</font> = <font color="Maroon" title="c15">x</font> as   <a href="osalg_1.html#NM1" title="OSALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S</font> ;<br/>
<a NAME="E9:69_1_1_2"/>
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K16" title="OSAFREE:func.16">FreeOSA</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="c16">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Maroon" title="c13">O</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">s</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c17">x1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c17">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K16" title="OSAFREE:func.16">FreeOSA</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="c16">s</font> or <font color="Maroon" title="c17">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Maroon" title="c13">O</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">s</font> )</span><br/>

<b>assume </b><a NAME="E1:69_1_1_2_3"/><i><font color="Green" title="E56">A54</font></i>: 
<font color="Maroon" title="c17">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K16" title="OSAFREE:func.16">FreeOSA</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="c16">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c17">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Maroon" title="c13">O</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">s</font></span><br/>

<a NAME="E2:69_1_1_2_3"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">NH1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K16" title="OSAFREE:func.16">FreeOSA</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="c15">x</font>
 
<b>by </b><i><a class="txt" href="#E3:69_1_1"><i><font color="Green" title="E46">A2</font></i></a>, <a class="txt" href="#E7:69_1_1_2"><i><font color="Green" title="E55">A53</font></i></a>, <a class="ref" href="msualg_3.html#D3" title="MSUALG_3:def.3">MSUALG_3:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c18">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:69_1_1_2_3"/><i><font color="Green" title="E57">A55</font></i>: 
( <font color="Maroon" title="c18">x2</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"><font color="Maroon" title="c8">NH1</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c16">s</font></span>)</span> &amp; <font color="Maroon" title="c17">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">NH1</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c16">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x2</font> )
 <b>by </b><i><a class="txt" href="#E1:69_1_1_2_3"><i><font color="Green" title="E56">A54</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:69_1_1_2_3"/><i><font color="Green" title="E58">A56</font></i>: 
<font color="Maroon" title="c18">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</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="c16">s</font>
 
<b>by </b><i><a class="txt" href="#E4:69_1_1_2_3"><i><font color="Green" title="E57">A55</font></i></a></i>;<br/>
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K8" title="OSAFREE:func.8">ParsedTermsOSA</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="c16">s</font> = 
 <a href="osafree.html#K4" title="OSAFREE:func.4">ParsedTerms</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c16">s</font>
<b>by </b><i><a class="ref" href="osafree.html#D8" target="_self" title="OSAFREE:def.8">Def8</a></i>
<br/>.= 
<span class="p1">{<span class="default"> <font color="Olive" title="b1">a</font> where <font color="Olive" title="b1">a</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> : (  ex <font color="Olive" title="b2">s2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> ex <font color="Olive" title="b3">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">s2</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c16">s</font> &amp; <font color="Olive" title="b3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">s2</font> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">x</font>,<font color="Olive" title="b2">s2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) or  ex <font color="Olive" title="b2">o1</font> being  <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">o1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Olive" title="b2">o1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c16">s</font> ) ) </span>}</span> 

;<br/>
<b>then consider </b><font color="Maroon" title="c19">a</font> being    <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> such that </b><br/><a NAME="E8:69_1_1_2_3"/><i><font color="Green" title="E59">A57</font></i>: 
<font color="Maroon" title="c19">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">x2</font>
 <b>and </b><br/><a NAME="E9:69_1_1_2_3"/>
(  ex <font color="Olive" title="b1">s2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> ex <font color="Olive" title="b2">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">s2</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c16">s</font> &amp; <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s2</font> &amp; <font color="Maroon" title="c19">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">s2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) or  ex <font color="Olive" title="b1">o1</font> being  <a href="osalg_1.html#NM2" title="OSALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">o1</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="osalg_1.html#K1" title="OSALG_1:func.1">the_result_sort_of</a> <font color="Olive" title="b1">o1</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c16">s</font> ) )
 <b>by </b><i><a class="txt" href="#E5:69_1_1_2_3"><i><font color="Green" title="E58">A56</font></i></a></i>;<br/>
<b>thus </b><a NAME="E10:69_1_1_2_3"/>
<font color="Maroon" title="c17">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Maroon" title="c13">O</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">s</font>
 <b>by </b><i><a class="txt" href="#E6:69_1_1_2"><i><font color="Green" title="E54">A52</font></i></a>, <a class="txt" href="#E4:69_1_1_2_3"><i><font color="Green" title="E57">A55</font></i></a>, <a class="txt" href="#E8:69_1_1_2_3"><i><font color="Green" title="E59">A57</font></i></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>hence </b><a NAME="E10:69_1_1_2"/>
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K16" title="OSAFREE:func.16">FreeOSA</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="c15">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Maroon" title="c13">O</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font>
 ; <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>
<a NAME="E18:69_1_1"/>
<font color="Maroon" title="c12">f</font> is    <a href="osafree.html#M1" title="OSAFREE:mode.1">OSGeneratorSet</a> of  <a href="osafree.html#K16" title="OSAFREE:func.16">FreeOSA</a> <font color="Maroon" title="c2">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">O1</font> be    <a href="osalg_2.html#M2" title="OSALG_2:mode.2">OSSubset</a> of  <a href="osafree.html#K16" title="OSAFREE:func.16">FreeOSA</a> <font color="Maroon" title="c2">X</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="osafree.html#D1" target="_self" title="OSAFREE:def.1">OSAFREE:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c14">O1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="osalg_2.html#K2" title="OSALG_2:func.2">OSCl</a> <font color="Maroon" title="c12">f</font> implies the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Maroon" title="c14">O1</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K16" title="OSAFREE:func.16">FreeOSA</a> <font color="Maroon" title="c2">X</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:69_1_1_3"/><i><font color="Green" title="E52">A58</font></i>: 
<font color="Maroon" title="c14">O1</font> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a>  <a href="osalg_2.html#K2" title="OSALG_2:func.2">OSCl</a> <font color="Maroon" title="c12">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Maroon" title="c14">O1</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K16" title="OSAFREE:func.16">FreeOSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span><br/>

<b>thus </b><a NAME="E2:69_1_1_3"/>
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osalg_2.html#K10" title="OSALG_2:func.10">GenOSAlg</a> <font color="Maroon" title="c14">O1</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="osafree.html#K16" title="OSAFREE:func.16">FreeOSA</a> <font color="Maroon" title="c2">X</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E17:69_1_1"><i><font color="Green" title="E51">A7</font></i></a>, <a class="txt" href="#E1:69_1_1_3"><i><font color="Green" title="E52">A58</font></i></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>then reconsider </b><font color="Maroon" title="c14">f</font> = <font color="Maroon" title="c12">f</font> as    <a href="osafree.html#M1" title="OSAFREE:mode.1">OSGeneratorSet</a> of  <a href="osafree.html#K16" title="OSAFREE:func.16">FreeOSA</a> <font color="Maroon" title="c2">X</font> ;<br/>
<b>take </b>
<font color="Maroon" title="c14">f</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">s</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> holds  <font color="Maroon" title="c14">f</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="osafree.html#K23" title="OSAFREE:func.23">OSFreeGen</a> <font color="Olive" title="b1">s</font>,<font color="Maroon" title="c2">X</font></span><br/>

<b>thus </b><a NAME="E20:69_1_1"/>
for <font color="Olive" title="b1">s</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">S</font> holds  <font color="Maroon" title="c14">f</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="osafree.html#K23" title="OSAFREE:func.23">OSFreeGen</a> <font color="Olive" title="b1">s</font>,<font color="Maroon" title="c2">X</font>
 <b>by </b><i><a class="txt" href="#E6:69_1_1"><i><font color="Green" title="E47">A3</font></i></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>
