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<span class="kw">let </span><font color="Maroon" title="c1">D9</font>, <font color="Maroon" title="c2">D</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="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">D9</font> holds <br/><font color="Olive" title="b1">x</font> is   <a href="trees_2.html#NM5" title="TREES_2:NM.5">DecoratedTree</a> of <font color="Maroon" title="c2">D</font> ) &amp; <font color="Maroon" title="c1">D9</font> is  <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a>  implies  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">D9</font> is   <a href="trees_2.html#NM5" title="TREES_2:NM.5">DecoratedTree</a> of <font color="Maroon" title="c2">D</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:69"/><span class="lab"><font color="Green" title="E33">A1</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">D9</font> holds <br/><font color="Olive" title="b1">x</font> is   <a href="trees_2.html#NM5" title="TREES_2:NM.5">DecoratedTree</a> of <font color="Maroon" title="c2">D</font>
 <span class="kw">and </span><br/><a NAME="E2:69"/><span class="lab"><font color="Green" title="E34">A2</font></span>: 
<font color="Maroon" title="c1">D9</font> is  <a href="ordinal1.html#V6" title="ORDINAL1:attr.6">c=-linear</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">D9</font> is   <a href="trees_2.html#NM5" title="TREES_2:NM.5">DecoratedTree</a> of <font color="Maroon" title="c2">D</font></span><br/>

<a NAME="E3:69"/><span class="lab"><font color="Green" title="E35">A3</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">D9</font> holds <br/><font color="Olive" title="b1">x</font> is   <a href="trees_2.html#NM4" title="TREES_2:NM.4">DecoratedTree</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69"><span class="lab"><font color="Green" title="E33">A1</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c3">T</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">D9</font> as   <a href="trees_2.html#NM4" title="TREES_2:NM.4">DecoratedTree</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:69"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="txt" href="#E3:69"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="ref" href="trees_2.html#T37" target="_self" title="TREES_2:th.37">Th37</a></span>;<br/>
<a NAME="E5:69"/><span class="lab"><font color="Green" title="E36">A4</font></span>: 
 <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> <font color="Maroon" title="c2">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1"><span class="kw">proof </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> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">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="c3">T</font> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:69_1"/><span class="lab"><font color="Green" title="E37">A5</font></span>: 
<font color="Maroon" title="c4">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="c3">T</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="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c5">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:69_1"/><span class="lab"><font color="Green" title="E38">A6</font></span>: 
( <font color="Maroon" title="c5">y</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> &amp; <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="c5">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_1"><span class="lab"><font color="Green" title="E37">A5</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E4:69_1"/><span class="lab"><font color="Green" title="E39">A7</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:69_1"><span class="lab"><font color="Green" title="E38">A6</font></span></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:69_1"/><span class="lab"><font color="Green" title="E40">A8</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font>
 <span class="kw">and </span><br/><a NAME="E7:69_1"/><span class="lab"><font color="Green" title="E41">A9</font></span>: 
<font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">D9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:69_1"><span class="lab"><font color="Green" title="E39">A7</font></span></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">X</font> = <font color="Maroon" title="c6">X</font> as   <a href="trees_2.html#NM5" title="TREES_2:NM.5">DecoratedTree</a> of <font color="Maroon" title="c2">D</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="txt" href="#E7:69_1"><span class="lab"><font color="Green" title="E41">A9</font></span></a></span>;<br/>
<a NAME="E9:69_1"/><span class="lab"><font color="Green" title="E42">A10</font></span>: 
( <font color="Maroon" title="c5">y</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="c7">X</font> &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">y</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:69_1"><span class="lab"><font color="Green" title="E40">A8</font></span></a>, <a class="ref" href="funct_1.html#T8" title="FUNCT_1:th.8">FUNCT_1:8</a></span>;<br/>
<a NAME="E10:69_1"/><span class="lab"><font color="Green" title="E43">A11</font></span>: 
<font color="Maroon" title="c4">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">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:69_1"><span class="lab"><font color="Green" title="E42">A10</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E11:69_1"/><span class="lab"><font color="Green" title="E44">A12</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">D</font>
 
<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/>
<span class="kw">thus </span><a NAME="E12:69_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:69_1"><span class="lab"><font color="Green" title="E43">A11</font></span></a>, <a class="txt" href="#E11:69_1"><span class="lab"><font color="Green" title="E44">A12</font></span></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><span class="kw">end;</span></div>
<span class="kw">thus </span><a NAME="E6:69"/>
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c1">D9</font> is   <a href="trees_2.html#NM5" title="TREES_2:NM.5">DecoratedTree</a> of <font color="Maroon" title="c2">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:69"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</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/>


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