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

<b>set </b><font color="Maroon" title="c3">fgs</font> =  <a href="freealg.html#K8" title="FREEALG:func.8">FreeGenSetNSG</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font>;<br/>
<b>set </b><font color="Maroon" title="c4">fua</font> =  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font>;<br/>
<a NAME="E1:31_1_1"/><i><font color="Green" title="E17">A1</font></i>: 
 <a href="freealg.html#K8" title="FREEALG:func.8">FreeGenSetNSG</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="freealg.html#T5" target="_self" title="FREEALG:th.5">Th5</a></i>;<br/>
<a NAME="E2:31_1_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K7" title="UNIALG_2:func.7">GenUnivAlg</a> <span class="p2">(<span class="default"><a href="freealg.html#K8" title="FREEALG:func.8">FreeGenSetNSG</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span></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 <span class="p1">(<span class="default"><a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:31_1_1_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K7" title="UNIALG_2:func.7">GenUnivAlg</a> <span class="p2">(<span class="default"><a href="freealg.html#K8" title="FREEALG:func.8">FreeGenSetNSG</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D8" title="UNIALG_2:def.8">UNIALG_2:def 8</a></i>;<br/>
<b>hence </b><a NAME="E2:31_1_1_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K7" title="UNIALG_2:func.7">GenUnivAlg</a> <span class="p2">(<span class="default"><a href="freealg.html#K8" title="FREEALG:func.8">FreeGenSetNSG</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <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 <span class="p1">(<span class="default"><a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> <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 <span class="p1">(<span class="default"><a href="unialg_2.html#K7" title="UNIALG_2:func.7">GenUnivAlg</a> <span class="p2">(<span class="default"><a href="freealg.html#K8" title="FREEALG:func.8">FreeGenSetNSG</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span></span><br/>

<b>set </b><font color="Maroon" title="c5">A</font> =  <a href="freealg.html#K3" title="FREEALG:func.3">DTConUA</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <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="freealg.html#K3" title="FREEALG:func.3">DTConUA</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span>] <b>means </b>$1 <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 <span class="p1">(<span class="default"><a href="unialg_2.html#K7" title="UNIALG_2:func.7">GenUnivAlg</a> <span class="p2">(<span class="default"><a href="freealg.html#K8" title="FREEALG:func.8">FreeGenSetNSG</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span>;<br/>
<a NAME="E3:31_1_1_1"/><i><font color="Green" title="E18">A2</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="freealg.html#K3" title="FREEALG:func.3">DTConUA</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</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="freealg.html#K3" title="FREEALG:func.3">DTConUA</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</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="31_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">s</font> be   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <span class="p1">(<span class="default"><a href="freealg.html#K3" title="FREEALG:func.3">DTConUA</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</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="c6">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="freealg.html#K3" title="FREEALG:func.3">DTConUA</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</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="c6">s</font>] )</span><br/>

<b>assume </b><a NAME="E1:31_1_1_1_1"/>
<font color="Maroon" title="c6">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="freealg.html#K3" title="FREEALG:func.3">DTConUA</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</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="c6">s</font>]</span><br/>

<a NAME="E2:31_1_1_1_1"/><b>then </b><i><font color="Green" title="E19">A3</font></i>: 
 <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Olive" title="b1">q</font></span>)</span> where <font color="Olive" title="b1">q</font> is   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <span class="p1">(<span class="default"><a href="freealg.html#K3" title="FREEALG:func.3">DTConUA</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> : <font color="Olive" title="b1">q</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="freealg.html#K3" title="FREEALG:func.3">DTConUA</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> </span>}</span> 
 
;<br/>
<a NAME="E3:31_1_1_1_1"/>
 <a href="freealg.html#K8" title="FREEALG:func.8">FreeGenSetNSG</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="freealg.html#T5" target="_self" title="FREEALG:th.5">Th5</a></i>;<br/>
<a NAME="E4:31_1_1_1_1"/><b>then </b>
 <a href="freealg.html#K8" title="FREEALG:func.8">FreeGenSetNSG</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</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 <span class="p1">(<span class="default"><a href="unialg_2.html#K7" title="UNIALG_2:func.7">GenUnivAlg</a> <span class="p2">(<span class="default"><a href="freealg.html#K8" title="FREEALG:func.8">FreeGenSetNSG</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D13" title="UNIALG_2:def.13">UNIALG_2:def 13</a></i>;<br/>
<b>hence </b><a NAME="E5:31_1_1_1_1"/>
<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="c6">s</font>]
 <b>by </b><i><a class="txt" href="freealg.html#E2:31_1_1_1_1"><i><font color="Green" title="E19">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><b>end;</b></div>
<a NAME="E4:31_1_1_1"/><i><font color="Green" title="E19">A4</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="freealg.html#K3" title="FREEALG:func.3">DTConUA</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</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="freealg.html#K3" title="FREEALG:func.3">DTConUA</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</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="freealg.html#K3" title="FREEALG:func.3">DTConUA</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</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="31_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">s</font> be   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <span class="p1">(<span class="default"><a href="freealg.html#K3" title="FREEALG:func.3">DTConUA</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</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="freealg.html#K3" title="FREEALG:func.3">DTConUA</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span>  st <font color="Maroon" title="c6">s</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="freealg.html#K3" title="FREEALG:func.3">DTConUA</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</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="c6">s</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="c7">p</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="freealg.html#K3" title="FREEALG:func.3">DTConUA</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</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="c6">s</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="c7">p</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="freealg.html#K3" title="FREEALG:func.3">DTConUA</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</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="c7">p</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="c6">s</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c7">p</font>] )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:31_1_1_1_2"/><i><font color="Green" title="E20">A5</font></i>: 
<font color="Maroon" title="c6">s</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="c7">p</font>
 <b>and </b><br/><a NAME="E2:31_1_1_1_2"/><i><font color="Green" title="E21">A6</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="freealg.html#K3" title="FREEALG:func.3">DTConUA</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</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="c7">p</font> holds <br/><font color="Olive" title="b1">t</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 <span class="p1">(<span class="default"><a href="unialg_2.html#K7" title="UNIALG_2:func.7">GenUnivAlg</a> <span class="p2">(<span class="default"><a href="freealg.html#K8" title="FREEALG:func.8">FreeGenSetNSG</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span></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>[<font color="Maroon" title="c6">s</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c7">p</font>]</span><br/>

<b>reconsider </b><font color="Maroon" title="c8">B</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="unialg_2.html#K7" title="UNIALG_2:func.7">GenUnivAlg</a> <span class="p2">(<span class="default"><a href="freealg.html#K8" title="FREEALG:func.8">FreeGenSetNSG</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> <b>by </b><i><a class="ref" href="unialg_2.html#D8" title="UNIALG_2:def.8">UNIALG_2:def 8</a></i>;<br/>
<a NAME="E4:31_1_1_1_2"/><i><font color="Green" title="E22">A7</font></i>: 
<font color="Maroon" title="c8">B</font> is <a href="unialg_2.html#V1" title="UNIALG_2:attr.1">opers_closed</a>
 
<b>by </b><i><a class="ref" href="unialg_2.html#D8" title="UNIALG_2:def.8">UNIALG_2:def 8</a></i>;<br/>
<a NAME="E5:31_1_1_1_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">p</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 <span class="p1">(<span class="default"><a href="unialg_2.html#K7" title="UNIALG_2:func.7">GenUnivAlg</a> <span class="p2">(<span class="default"><a href="freealg.html#K8" title="FREEALG:func.8">FreeGenSetNSG</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">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="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="c9">x</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="c7">p</font> or <font color="Maroon" title="c9">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 <span class="p1">(<span class="default"><a href="unialg_2.html#K7" title="UNIALG_2:func.7">GenUnivAlg</a> <span class="p2">(<span class="default"><a href="freealg.html#K8" title="FREEALG:func.8">FreeGenSetNSG</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:31_1_1_1_2_1"/><i><font color="Green" title="E23">A8</font></i>: 
<font color="Maroon" title="c9">x</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="c7">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">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 <span class="p1">(<span class="default"><a href="unialg_2.html#K7" title="UNIALG_2:func.7">GenUnivAlg</a> <span class="p2">(<span class="default"><a href="freealg.html#K8" title="FREEALG:func.8">FreeGenSetNSG</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span></span><br/>

<a NAME="E2:31_1_1_1_2_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="trees_3.html#K5" title="TREES_3:func.5">FinTrees</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="freealg.html#K3" title="FREEALG:func.3">DTConUA</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</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/>
<b>then reconsider </b><font color="Maroon" title="c10">x1</font> = <font color="Maroon" title="c9">x</font> as    <a href="trees_3.html#M5" title="TREES_3:mode.5">Element</a> of  <a href="trees_3.html#K5" title="TREES_3:func.5">FinTrees</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="freealg.html#K3" title="FREEALG:func.3">DTConUA</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> <b>by </b><i><a class="txt" href="freealg.html#E1:31_1_1_1_2_1"><i><font color="Green" title="E23">A8</font></i></a></i>;<br/>
<a NAME="E4:31_1_1_1_2_1"/>
<font color="Maroon" title="c10">x1</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 <span class="p1">(<span class="default"><a href="unialg_2.html#K7" title="UNIALG_2:func.7">GenUnivAlg</a> <span class="p2">(<span class="default"><a href="freealg.html#K8" title="FREEALG:func.8">FreeGenSetNSG</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="freealg.html#E2:31_1_1_1_2"><i><font color="Green" title="E21">A6</font></i></a>, <a class="txt" href="freealg.html#E1:31_1_1_1_2_1"><i><font color="Green" title="E23">A8</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:31_1_1_1_2_1"/>
<font color="Maroon" title="c9">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 <span class="p1">(<span class="default"><a href="unialg_2.html#K7" title="UNIALG_2:func.7">GenUnivAlg</a> <span class="p2">(<span class="default"><a href="freealg.html#K8" title="FREEALG:func.8">FreeGenSetNSG</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span>
 ; <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="c9">cp</font> = <font color="Maroon" title="c7">p</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</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="unialg_2.html#K7" title="UNIALG_2:func.7">GenUnivAlg</a> <span class="p2">(<span class="default"><a href="freealg.html#K8" title="FREEALG:func.8">FreeGenSetNSG</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span></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="E7:31_1_1_1_2"/><i><font color="Green" title="E23">A9</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">s</font>,<span class="p2">(<span class="default"><a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c7">p</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="freealg.html#K3" title="FREEALG:func.3">DTConUA</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> &amp; the <a href="lang1.html#U1" title="LANG1:sel.1">Rules</a> of <span class="p1">(<span class="default"><a href="freealg.html#K3" title="FREEALG:func.3">DTConUA</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="freealg.html#K2" title="FREEALG:func.2">REL</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font> )
 
<b>by </b><i><a class="txt" href="freealg.html#E1:31_1_1_1_2"><i><font color="Green" title="E20">A5</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="c10">s0</font> = <font color="Maroon" title="c6">s</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">D</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c11">rp</font> =  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c7">p</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">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">f</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <b>by </b><i><a class="txt" href="freealg.html#E7:31_1_1_1_2"><i><font color="Green" title="E23">A9</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E10:31_1_1_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">s0</font>,<font color="Maroon" title="c11">rp</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="freealg.html#K2" title="FREEALG:func.2">REL</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font>
 
<b>by </b><i><a class="txt" href="freealg.html#E1:31_1_1_1_2"><i><font color="Green" title="E20">A5</font></i></a>, <a class="ref" href="lang1.html#D1" title="LANG1:def.1">LANG1:def 1</a></i>;<br/>
<a NAME="E11:31_1_1_1_2"/><b>then </b><i><font color="Green" title="E24">A10</font></i>: 
( <font color="Maroon" title="c10">s0</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="c1">f</font> &amp; <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">s0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">rp</font> )
 
<b>by </b><i><a class="ref" href="freealg.html#D8" target="_self" title="FREEALG:def.8">Def8</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c12">ns</font> = <font color="Maroon" title="c6">s</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E13:31_1_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="freealg.html#K6" title="FREEALG:func.6">FreeOpSeqNSG</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">f</font>
 
<b>by </b><i><a class="ref" href="freealg.html#D12" target="_self" title="FREEALG:def.12">Def12</a></i>;<br/>
<b>then </b> <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="freealg.html#K6" title="FREEALG:func.6">FreeOpSeqNSG</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> = 
 <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="c1">f</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 href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">f</font>
<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="E15:31_1_1_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="freealg.html#K6" title="FREEALG:func.6">FreeOpSeqNSG</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">ns</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> <span class="p1">(<span class="default"><a href="freealg.html#K6" title="FREEALG:func.6">FreeOpSeqNSG</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="freealg.html#K6" title="FREEALG:func.6">FreeOpSeqNSG</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">ns</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="freealg.html#K5" title="FREEALG:func.5">FreeOpNSG</a> <font color="Maroon" title="c12">ns</font>,<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> the <a href="unialg_1.html#U1" title="UNIALG_1:sel.1">charact</a> of <span class="p1">(<span class="default"><a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="unialg_2.html#K1" title="UNIALG_2:func.1">Operations</a> <span class="p1">(<span class="default"><a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="freealg.html#E11:31_1_1_1_2"><i><font color="Green" title="E24">A10</font></i></a>, <a class="ref" href="freealg.html#D12" target="_self" title="FREEALG:def.12">Def12</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="c13">o</font> =  <a href="freealg.html#K5" title="FREEALG:func.5">FreeOpNSG</a> <font color="Maroon" title="c12">ns</font>,<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font> as   <a href="unialg_2.html#NM5" title="UNIALG_2:NM.5">operation</a> of <span class="p1">(<span class="default"><a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> ;<br/>
<a NAME="E17:31_1_1_1_2"/><i><font color="Green" title="E25">A11</font></i>: 
<font color="Maroon" title="c8">B</font> <a href="unialg_2.html#R2" title="UNIALG_2:pred.2">is_closed_on</a> <font color="Maroon" title="c13">o</font>
 
<b>by </b><i><a class="txt" href="freealg.html#E4:31_1_1_1_2"><i><font color="Green" title="E22">A7</font></i></a>, <a class="ref" href="unialg_2.html#D5" title="UNIALG_2:def.5">UNIALG_2:def 5</a></i>;<br/>
<a NAME="E18:31_1_1_1_2"/><i><font color="Green" title="E26">A12</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="freealg.html#K5" title="FREEALG:func.5">FreeOpNSG</a> <font color="Maroon" title="c12">ns</font>,<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">ns</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p2">(<span class="default"><a href="freealg.html#K3" title="FREEALG:func.3">DTConUA</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="freealg.html#E11:31_1_1_1_2"><i><font color="Green" title="E24">A10</font></i></a>, <a class="ref" href="freealg.html#D11" target="_self" title="FREEALG:def.11">Def11</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">tp</font> = <font color="Maroon" title="c7">p</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p2">(<span class="default"><a href="freealg.html#K3" title="FREEALG:func.3">DTConUA</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></i>;<br/>
 <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="c7">p</font></span>)</span> = 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">p</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 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="c7">p</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/>
<b>then </b><i><font color="Green" title="E27">A13</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">p</font> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">rp</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">ns</font>
<b>by </b><i><a class="txt" href="freealg.html#E11:31_1_1_1_2"><i><font color="Green" title="E24">A10</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
;<br/>
<a NAME="E22:31_1_1_1_2"/><b>then </b><i><font color="Green" title="E28">A14</font></i>: 
<font color="Maroon" title="c14">tp</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p2">(<span class="default"><a href="freealg.html#K3" title="FREEALG:func.3">DTConUA</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  :  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">ns</font> </span>}</span> 
 
;<br/>
<a NAME="E23:31_1_1_1_2"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c13">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c13">o</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span>
 
<b>by </b><i><a class="ref" href="unialg_2.html#T2" title="UNIALG_2:th.2">UNIALG_2:2</a></i>;<br/>
<a NAME="E24:31_1_1_1_2"/><b>then </b><i><font color="Green" title="E29">A15</font></i>: 
 <a href="unialg_1.html#K3" title="UNIALG_1:func.3">arity</a> <font color="Maroon" title="c13">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">ns</font>
 
<b>by </b><i><a class="txt" href="freealg.html#E18:31_1_1_1_2"><i><font color="Green" title="E26">A12</font></i></a>, <a class="ref" href="pralg_1.html#T1" title="PRALG_1:th.1">PRALG_1:1</a></i>;<br/>
<font color="Maroon" title="c13">o</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">cp</font> = 
<span class="p1">(<span class="default"><a href="freealg.html#K4" title="FREEALG:func.4">Sym</a> <font color="Maroon" title="c12">ns</font>,<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c7">p</font>
<b>by </b><i><a class="txt" href="freealg.html#E11:31_1_1_1_2"><i><font color="Green" title="E24">A10</font></i></a>, <a class="txt" href="freealg.html#E18:31_1_1_1_2"><i><font color="Green" title="E26">A12</font></i></a>, <a class="txt" href="freealg.html#E22:31_1_1_1_2"><i><font color="Green" title="E28">A14</font></i></a>, <a class="ref" href="freealg.html#D11" target="_self" title="FREEALG:def.11">Def11</a></i>
<br/>.= 
<font color="Maroon" title="c6">s</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c7">p</font>
<b>by </b><i><a class="txt" href="freealg.html#E11:31_1_1_1_2"><i><font color="Green" title="E24">A10</font></i></a>, <a class="ref" href="freealg.html#D10" target="_self" title="FREEALG:def.10">Def10</a></i>
;<br/>
<b>hence </b><a NAME="E26:31_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">s</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c7">p</font>]
 <b>by </b><i><a class="txt" href="freealg.html#E17:31_1_1_1_2"><i><font color="Green" title="E25">A11</font></i></a>, <a class="txt" href="freealg.html#E21:31_1_1_1_2"><i><font color="Green" title="E27">A13</font></i></a>, <a class="txt" href="freealg.html#E24:31_1_1_1_2"><i><font color="Green" title="E29">A15</font></i></a>, <a class="ref" href="unialg_2.html#D4" title="UNIALG_2:def.4">UNIALG_2:def 4</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="E5:31_1_1_1"/><i><font color="Green" title="E20">A16</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="freealg.html#K3" title="FREEALG:func.3">DTConUA</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</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="freealg.html#K3" title="FREEALG:func.3">DTConUA</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</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="freealg.html#E3:31_1_1_1"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="freealg.html#E4:31_1_1_1"><i><font color="Green" title="E19">A4</font></i></a>);<br/></i>
<b>let </b><font color="Maroon" title="c6">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="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="c6">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 <span class="p1">(<span class="default"><a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> or <font color="Maroon" title="c6">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 <span class="p1">(<span class="default"><a href="unialg_2.html#K7" title="UNIALG_2:func.7">GenUnivAlg</a> <span class="p2">(<span class="default"><a href="freealg.html#K8" title="FREEALG:func.8">FreeGenSetNSG</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span> )</span><br/>

<b>assume </b><a NAME="E6:31_1_1_1"/><i><font color="Green" title="E21">A17</font></i>: 
<font color="Maroon" title="c6">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 <span class="p1">(<span class="default"><a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</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="c6">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 <span class="p1">(<span class="default"><a href="unialg_2.html#K7" title="UNIALG_2:func.7">GenUnivAlg</a> <span class="p2">(<span class="default"><a href="freealg.html#K8" title="FREEALG:func.8">FreeGenSetNSG</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span></span><br/>

<b>then reconsider </b><font color="Maroon" title="c7">x1</font> = <font color="Maroon" title="c6">x</font> as    <a href="trees_3.html#M5" title="TREES_3:mode.5">Element</a> of  <a href="trees_3.html#K5" title="TREES_3:func.5">FinTrees</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="freealg.html#K3" title="FREEALG:func.3">DTConUA</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span> ;<br/>
<a NAME="E8:31_1_1_1"/>
<font color="Maroon" title="c7">x1</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="freealg.html#K3" title="FREEALG:func.3">DTConUA</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span>
 
<b>by </b><i><a class="txt" href="freealg.html#E6:31_1_1_1"><i><font color="Green" title="E21">A17</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:31_1_1_1"/>
<font color="Maroon" title="c6">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 <span class="p1">(<span class="default"><a href="unialg_2.html#K7" title="UNIALG_2:func.7">GenUnivAlg</a> <span class="p2">(<span class="default"><a href="freealg.html#K8" title="FREEALG:func.8">FreeGenSetNSG</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="freealg.html#E5:31_1_1_1"><i><font color="Green" title="E20">A16</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="E3:31_1_1"/>
 <a href="freealg.html#K8" title="FREEALG:func.8">FreeGenSetNSG</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font> is    <a href="freealg.html#M1" title="FREEALG:mode.1">GeneratorSet</a> of  <a href="freealg.html#K7" title="FREEALG:func.7">FreeUnivAlgNSG</a> <font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">D</font>
 <b>by </b><i><a class="txt" href="freealg.html#E1:31_1_1"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="freealg.html#E8"><i><font color="Green" title="E6">Lm3</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>
