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

<span class="kw">let </span><font color="Maroon" title="c1">S</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  non  <a href="struct_0.html#V11" title="STRUCT_0:attr.11">void</a>   <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">V</font> being <a href="relat_1.html#V2" title="RELAT_1:attr.2">V5</a>() <a href="pboole.html#NM1" title="PBOOLE:NM.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><br/> for <font color="Olive" title="b2">X1</font>, <font color="Olive" title="b3">X2</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="msaterm.html#K1" title="MSATERM:func.1">-Terms</a> <font color="Olive" title="b1">V</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Olive" title="b2">X1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b3">X2</font></span>)</span> <a href="circtrm1.html#K1" title="CIRCTRM1:func.1">-CircuitStr</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">X1</font> <a href="circtrm1.html#K1" title="CIRCTRM1:func.1">-CircuitStr</a> </span>)</span> <a href="circcomb.html#K2" title="CIRCCOMB:func.2">+*</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">X2</font> <a href="circtrm1.html#K1" title="CIRCTRM1:func.1">-CircuitStr</a> </span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">V</font> be  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V5</a>() <a href="pboole.html#NM1" title="PBOOLE:NM.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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">X1</font>, <font color="Olive" title="b2">X2</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="msaterm.html#K1" title="MSATERM:func.1">-Terms</a> <font color="Maroon" title="c2">V</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">X1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">X2</font></span>)</span> <a href="circtrm1.html#K1" title="CIRCTRM1:func.1">-CircuitStr</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">X1</font> <a href="circtrm1.html#K1" title="CIRCTRM1:func.1">-CircuitStr</a> </span>)</span> <a href="circcomb.html#K2" title="CIRCCOMB:func.2">+*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">X2</font> <a href="circtrm1.html#K1" title="CIRCTRM1:func.1">-CircuitStr</a> </span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">X1</font>, <font color="Maroon" title="c4">X2</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="msaterm.html#K1" title="MSATERM:func.1">-Terms</a> <font color="Maroon" title="c2">V</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">X2</font></span>)</span> <a href="circtrm1.html#K1" title="CIRCTRM1:func.1">-CircuitStr</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="circtrm1.html#K1" title="CIRCTRM1:func.1">-CircuitStr</a> </span>)</span> <a href="circcomb.html#K2" title="CIRCCOMB:func.2">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="circtrm1.html#K1" title="CIRCTRM1:func.1">-CircuitStr</a> </span>)</span></span><br/>





<span class="kw">set </span><font color="Maroon" title="c5">X</font> = <font color="Maroon" title="c3">X1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">X2</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">C</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>;<br/>
<a NAME="E1:19"/><span class="lab"><font color="Green" title="E10">A1</font></span>: 
 <a href="trees_9.html#K9" title="TREES_9:func.9">Subtrees</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">X2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="trees_9.html#K9" title="TREES_9:func.9">Subtrees</a> <font color="Maroon" title="c3">X1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="trees_9.html#K9" title="TREES_9:func.9">Subtrees</a> <font color="Maroon" title="c4">X2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="circtrm1.html#T8" target="_self" title="CIRCTRM1:th.8">Th8</a></span>;<br/>
<a NAME="E2:19"/><span class="lab"><font color="Green" title="E11">A2</font></span>: 
<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="trees_9.html#K12" title="TREES_9:func.12">-Subtrees</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">X2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="trees_9.html#K12" title="TREES_9:func.12">-Subtrees</a> <font color="Maroon" title="c3">X1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <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="trees_9.html#K12" title="TREES_9:func.12">-Subtrees</a> <font color="Maroon" title="c4">X2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="circtrm1.html#T9" target="_self" title="CIRCTRM1:th.9">Th9</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><span class="kw">now </span></a><div class="add"><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">t</font> being   <a href="trees_3.html#M3" title="TREES_3:mode.3">Element</a> of <font color="Maroon" title="c4">X2</font> holds  <font color="Olive" title="b1">t</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c7">t</font> be    <a href="trees_3.html#M3" title="TREES_3:mode.3">Element</a> of <font color="Maroon" title="c3">X1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">t</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span><br/><a NAME="E1:19_1_1"/>
<font color="Maroon" title="c7">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X1</font>
 ;<br/><span class="kw">hence </span><a NAME="E2:19_1_1"/>
<font color="Maroon" title="c7">t</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c7">t</font> be    <a href="trees_3.html#M3" title="TREES_3:mode.3">Element</a> of <font color="Maroon" title="c4">X2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">t</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span><br/><a NAME="E1:19_1_2"/>
<font color="Maroon" title="c7">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X2</font>
 ;<br/><span class="kw">hence </span><a NAME="E2:19_1_2"/>
<font color="Maroon" title="c7">t</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E4:19"/><span class="kw">then </span><span class="lab"><font color="Green" title="E12">A3</font></span>: 
<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="trees_9.html#K13" title="TREES_9:func.13">-ImmediateSubtrees</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">X2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="trees_9.html#K13" title="TREES_9:func.13">-ImmediateSubtrees</a> <font color="Maroon" title="c3">X1</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <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="trees_9.html#K13" title="TREES_9:func.13">-ImmediateSubtrees</a> <font color="Maroon" title="c4">X2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="circtrm1.html#T10" target="_self" title="CIRCTRM1:th.10">Th10</a></span>;<br/>
<a NAME="E5:19"/>
 <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <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="trees_9.html#K12" title="TREES_9:func.12">-Subtrees</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">X2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default"><span class="p3"><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="p4"><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="trees_9.html#K12" title="TREES_9:func.12">-Subtrees</a> <font color="Maroon" title="c3">X1</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default"><span class="p3"><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="p4"><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="trees_9.html#K12" title="TREES_9:func.12">-Subtrees</a> <font color="Maroon" title="c4">X2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:19"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="ref" href="funct_4.html#T23" title="FUNCT_4:th.23">FUNCT_4:23</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:19"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c4">X2</font></span>)</span> <a href="circtrm1.html#K1" title="CIRCTRM1:func.1">-CircuitStr</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">X1</font> <a href="circtrm1.html#K1" title="CIRCTRM1:func.1">-CircuitStr</a> </span>)</span> <a href="circcomb.html#K2" title="CIRCCOMB:func.2">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">X2</font> <a href="circtrm1.html#K1" title="CIRCTRM1:func.1">-CircuitStr</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="txt" href="#E2:19"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="txt" href="#E4:19"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="ref" href="circcomb.html#D2" title="CIRCCOMB:def.2">CIRCCOMB:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
