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<b>let </b><font color="Maroon" title="c1">D1</font>, <font color="Maroon" title="c2">D2</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <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"> for <font color="Olive" title="b1">T</font> being  <a href="trees_3.html#NM7" title="TREES_3:NM.7">DecoratedTree</a> of <font color="Maroon" title="c1">D1</font>,<font color="Maroon" title="c2">D2</font> holds  <span class="p1"><a href="trees_3.html#K6" title="TREES_3:func.6">&lt;:</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">T</font> <a href="trees_3.html#K10" title="TREES_3:func.10">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">T</font> <a href="trees_3.html#K11" title="TREES_3:func.11">`2</a> </span>)</span></span><a href="trees_3.html#K6" title="TREES_3:func.6">:&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">T</font></span><br/><b>let </b><font color="Maroon" title="c3">T</font> be   <a href="trees_3.html#NM7" title="TREES_3:NM.7">DecoratedTree</a> of <font color="Maroon" title="c1">D1</font>,<font color="Maroon" title="c2">D2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="trees_3.html#K6" title="TREES_3:func.6">&lt;:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="trees_3.html#K10" title="TREES_3:func.10">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="trees_3.html#K11" title="TREES_3:func.11">`2</a> </span>)</span></span><a href="trees_3.html#K6" title="TREES_3:func.6">:&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">T</font></span><br/>



<a NAME="E1:94"/>
( <font color="Maroon" title="c3">T</font> <a href="trees_3.html#K10" title="TREES_3:func.10">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="trees_3.html#K8" title="TREES_3:func.8">pr1</a> <font color="Maroon" title="c1">D1</font>,<font color="Maroon" title="c2">D2</font></span>)</span> <a href="trees_3.html#K7" title="TREES_3:func.7">*</a> <font color="Maroon" title="c3">T</font> &amp; <font color="Maroon" title="c3">T</font> <a href="trees_3.html#K11" title="TREES_3:func.11">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="trees_3.html#K9" title="TREES_3:func.9">pr2</a> <font color="Maroon" title="c1">D1</font>,<font color="Maroon" title="c2">D2</font></span>)</span> <a href="trees_3.html#K7" title="TREES_3:func.7">*</a> <font color="Maroon" title="c3">T</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="trees_3.html#K8" title="TREES_3:func.8">pr1</a> <font color="Maroon" title="c1">D1</font>,<font color="Maroon" title="c2">D2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">D1</font>,<font color="Maroon" title="c2">D2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="trees_3.html#K9" title="TREES_3:func.9">pr2</a> <font color="Maroon" title="c1">D1</font>,<font color="Maroon" title="c2">D2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">D1</font>,<font color="Maroon" title="c2">D2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">D1</font>,<font color="Maroon" title="c2">D2</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 
<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="trees_2.html#D9" title="TREES_2:def.9">TREES_2:def 9</a></i>;<br/>
<a NAME="E2:94"/><b>then </b><i><font color="Green" title="E42">A1</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="trees_3.html#K10" title="TREES_3:func.10">`1</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">T</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="trees_3.html#K11" title="TREES_3:func.11">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">T</font> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/>
<a NAME="E3:94"/><b>then </b><i><font color="Green" title="E43">A2</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="trees_3.html#K6" title="TREES_3:func.6">&lt;:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="trees_3.html#K10" title="TREES_3:func.10">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="trees_3.html#K11" title="TREES_3:func.11">`2</a> </span>)</span></span><a href="trees_3.html#K6" title="TREES_3:func.6">:&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">T</font>
 
<b>by </b><i><a class="ref" href="funct_3.html#T70" title="FUNCT_3:th.70">FUNCT_3:70</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="94_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">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="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">T</font> implies <span class="p1"><a href="trees_3.html#K6" title="TREES_3:func.6">&lt;:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="trees_3.html#K10" title="TREES_3:func.10">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="trees_3.html#K11" title="TREES_3:func.11">`2</a> </span>)</span></span><a href="trees_3.html#K6" title="TREES_3:func.6">:&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> )</span><br/><b>assume </b><a NAME="E1:94_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">T</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="trees_3.html#K6" title="TREES_3:func.6">&lt;:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="trees_3.html#K10" title="TREES_3:func.10">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="trees_3.html#K11" title="TREES_3:func.11">`2</a> </span>)</span></span><a href="trees_3.html#K6" title="TREES_3:func.6">:&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font></span><br/><b>then reconsider </b><font color="Maroon" title="c5">t</font> = <font color="Maroon" title="c4">x</font> as    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">T</font> ;<br/><b>thus </b><span class="p1"><a href="trees_3.html#K6" title="TREES_3:func.6">&lt;:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="trees_3.html#K10" title="TREES_3:func.10">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="trees_3.html#K11" title="TREES_3:func.11">`2</a> </span>)</span></span><a href="trees_3.html#K6" title="TREES_3:func.6">:&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font> = 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="trees_3.html#K10" title="TREES_3:func.10">`1</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">t</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="trees_3.html#K11" title="TREES_3:func.11">`2</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="txt" href="trees_3.html#E2:94"><i><font color="Green" title="E42">A1</font></i></a>, <a class="ref" href="funct_3.html#T69" title="FUNCT_3:th.69">FUNCT_3:69</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c5">t</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="trees_3.html#K11" title="TREES_3:func.11">`2</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">t</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<b>by </b><i><a class="ref" href="trees_3.html#T41" target="_self" title="TREES_3:th.41">Th41</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c5">t</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c5">t</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
<b>by </b><i><a class="ref" href="trees_3.html#T41" target="_self" title="TREES_3:th.41">Th41</a></i>
<br/>.= 
<font color="Maroon" title="c3">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x</font>
<b>by </b><i><a class="ref" href="mcart_1.html#T23" title="MCART_1:th.23">MCART_1:23</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="E5:94"/>
<span class="p1"><a href="trees_3.html#K6" title="TREES_3:func.6">&lt;:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="trees_3.html#K10" title="TREES_3:func.10">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="trees_3.html#K11" title="TREES_3:func.11">`2</a> </span>)</span></span><a href="trees_3.html#K6" title="TREES_3:func.6">:&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">T</font>
 <b>by </b><i><a class="txt" href="trees_3.html#E3:94"><i><font color="Green" title="E43">A2</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</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>
