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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><br/> for <font color="Olive" title="b2">t</font> being   <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="Olive" title="b1">T</font> holds <br/> ( <span class="p1">(<span class="default"><font color="Olive" title="b1">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Olive" title="b2">t</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<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> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">t</font> &amp; <span class="p1">(<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> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Olive" title="b2">t</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  )</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"> for <font color="Olive" title="b1">t</font> being   <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> holds <br/> ( <span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Olive" title="b1">t</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">t</font> &amp; <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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Olive" title="b1">t</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  )</span><br/><b>let </b><font color="Maroon" title="c4">t</font> be    <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>; <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="c3">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c4">t</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">t</font> &amp; <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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<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="c4">t</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  )</span><br/>





<a NAME="E1:93"/>
( <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="E2:93"/><b>then </b><i><font color="Green" title="E40">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="E3:93"/>
<font color="Maroon" title="c3">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c4">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c4">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="c4">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="mcart_1.html#T23" title="MCART_1:th.23">MCART_1:23</a></i>;<br/>
<a NAME="E4:93"/><b>then </b>
( <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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">t</font> <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="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<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="c4">t</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<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="c4">t</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> &amp; <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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">t</font> <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="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><span class="p2">(<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="c4">t</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<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="c4">t</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span> )
 
<b>by </b><i><a class="txt" href="#E2:93"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/>
<b>hence </b><a NAME="E5:93"/>
( <span class="p1">(<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="c4">t</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">t</font> &amp; <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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<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="c4">t</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  )
 <b>by </b><i><a class="ref" href="funct_3.html#D5" title="FUNCT_3:def.5">FUNCT_3:def 5</a>, <a class="ref" href="funct_3.html#D6" title="FUNCT_3:def.6">FUNCT_3:def 6</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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