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<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">T</font> being   <a href="trees_3.html#NM2" title="TREES_3:NM.2">DecoratedTree</a> of <font color="Maroon" title="c1">D1</font>,<font color="Maroon" title="c2">D2</font> holds  <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">T</font> <a href="trees_3.html#K8" title="TREES_3:func.8">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">T</font> <a href="trees_3.html#K9" title="TREES_3:func.9">`2</a></span>)</span></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">T</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">T</font> be   <a href="trees_3.html#NM2" title="TREES_3:NM.2">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="trees_3.html#K8" title="TREES_3:func.8">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="trees_3.html#K9" title="TREES_3:func.9">`2</a></span>)</span></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">T</font></span><br/>



<a NAME="E1:93"/><span class="lab"><font color="Green" title="E39">A1</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="trees_3.html#K6" title="TREES_3:func.6">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E2:93"/><span class="lab"><font color="Green" title="E40">A2</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="trees_3.html#K7" title="TREES_3:func.7">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E3:93"/><span class="lab"><font color="Green" title="E41">A3</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">T</font> <a href="relset_1.html#R1" title="RELSET_1: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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/>
<a NAME="E4:93"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A4</font></span>: 
 <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#K8" title="TREES_3:func.8">`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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:93"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="relat_1.html#T27" title="RELAT_1:th.27">RELAT_1:27</a></span>;<br/>
<a NAME="E5:93"/><span class="lab"><font color="Green" title="E43">A5</font></span>: 
 <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#K9" title="TREES_3:func.9">`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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:93"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="txt" href="#E3:93"><span class="lab"><font color="Green" title="E41">A3</font></span></a>, <a class="ref" href="relat_1.html#T27" title="RELAT_1:th.27">RELAT_1:27</a></span>;<br/>
<a NAME="E6:93"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A6</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="trees_3.html#K8" title="TREES_3:func.8">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="trees_3.html#K9" title="TREES_3:func.9">`2</a></span>)</span></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:93"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <a class="ref" href="funct_3.html#T50" title="FUNCT_3:th.50">FUNCT_3:50</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="93_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="trees_3.html#K8" title="TREES_3:func.8">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="trees_3.html#K9" title="TREES_3:func.9">`2</a></span>)</span></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&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/><span class="kw">assume </span><a NAME="E1:93_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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="trees_3.html#K8" title="TREES_3:func.8">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="trees_3.html#K9" title="TREES_3:func.9">`2</a></span>)</span></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&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/><span class="kw">then </span><span class="kw">reconsider </span><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/><span class="kw">thus </span><span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="trees_3.html#K8" title="TREES_3:func.8">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="trees_3.html#K9" title="TREES_3:func.9">`2</a></span>)</span></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&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#K8" title="TREES_3:func.8">`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#K9" title="TREES_3:func.9">`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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:93"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <a class="txt" href="#E5:93"><span class="lab"><font color="Green" title="E43">A5</font></span></a>, <a class="ref" href="funct_3.html#T49" title="FUNCT_3:th.49">FUNCT_3:49</a></span>
<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#K9" title="TREES_3:func.9">`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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="trees_3.html#T39" target="_self" title="TREES_3:th.39">Th41</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="trees_3.html#T39" target="_self" title="TREES_3:th.39">Th41</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T21" title="MCART_1:th.21">MCART_1:21</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E8:93"/>
<span class="p1"><a href="funct_3.html#K13" title="FUNCT_3:func.13">&lt;:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="trees_3.html#K8" title="TREES_3:func.8">`1</a></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="trees_3.html#K9" title="TREES_3:func.9">`2</a></span>)</span></span><a href="funct_3.html#K13" title="FUNCT_3:func.13">:&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">T</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:93"><span class="lab"><font color="Green" title="E44">A6</font></span></a>, <a class="ref" href="funct_1.html#T2" title="FUNCT_1:th.2">FUNCT_1:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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