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<b>let </b><font color="Maroon" title="c2">n1</font>, <font color="Maroon" title="c3">n2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  ex <font color="Olive" title="b1">X</font> being   <a href="trees_1.html#M4" title="TREES_1:mode.4">AntiChain_of_Prefixes</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Maroon" title="c2">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">X</font> &amp; ( for <font color="Olive" title="b2">Y</font> being   <a href="trees_1.html#M4" title="TREES_1:mode.4">AntiChain_of_Prefixes</a> of <font color="Maroon" title="c1">T</font> holds   <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b2">Y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">X</font> ) ) &amp;  ex <font color="Olive" title="b1">X</font> being   <a href="trees_1.html#M4" title="TREES_1:mode.4">AntiChain_of_Prefixes</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Maroon" title="c3">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">X</font> &amp; ( for <font color="Olive" title="b2">Y</font> being   <a href="trees_1.html#M4" title="TREES_1:mode.4">AntiChain_of_Prefixes</a> of <font color="Maroon" title="c1">T</font> holds   <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b2">Y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">X</font> ) ) implies <font color="Maroon" title="c2">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n2</font> )</span><br/>

<b>given </b><font color="Maroon" title="c4">X1</font> being    <a href="trees_1.html#M4" title="TREES_1:mode.4">AntiChain_of_Prefixes</a> of <font color="Maroon" title="c1">T</font><b> such that </b><a NAME="E2:65_2_2"/><i><font color="Green" title="E40">A22</font></i>: 
( <font color="Maroon" title="c2">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">X1</font> &amp; ( for <font color="Olive" title="b1">Y</font> being   <a href="trees_1.html#M4" title="TREES_1:mode.4">AntiChain_of_Prefixes</a> of <font color="Maroon" title="c1">T</font> holds   <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">Y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">X1</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">X</font> being   <a href="trees_1.html#M4" title="TREES_1:mode.4">AntiChain_of_Prefixes</a> of <font color="Maroon" title="c1">T</font>  holds <br/>(  not <font color="Maroon" title="c3">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">X</font> or  ex <font color="Olive" title="b2">Y</font> being   <a href="trees_1.html#M4" title="TREES_1:mode.4">AntiChain_of_Prefixes</a> of <font color="Maroon" title="c1">T</font> st not  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b2">Y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">X</font> ) or <font color="Maroon" title="c2">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n2</font> )</span><br/>

<b>given </b><font color="Maroon" title="c5">X2</font> being    <a href="trees_1.html#M4" title="TREES_1:mode.4">AntiChain_of_Prefixes</a> of <font color="Maroon" title="c1">T</font><b> such that </b><a NAME="E4:65_2_2"/><i><font color="Green" title="E41">A23</font></i>: 
( <font color="Maroon" title="c3">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">X2</font> &amp; ( for <font color="Olive" title="b1">Y</font> being   <a href="trees_1.html#M4" title="TREES_1:mode.4">AntiChain_of_Prefixes</a> of <font color="Maroon" title="c1">T</font> holds   <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Olive" title="b1">Y</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">X2</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n2</font></span><br/>

<a NAME="E5:65_2_2"/>
(  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">X1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">X2</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">X2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c4">X1</font> )
 
<b>by </b><i><a class="txt" href="#E2:65_2_2"><i><font color="Green" title="E40">A22</font></i></a>, <a class="txt" href="#E4:65_2_2"><i><font color="Green" title="E41">A23</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:65_2_2"/>
<font color="Maroon" title="c2">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">n2</font>
 <b>by </b><i><a class="txt" href="#E2:65_2_2"><i><font color="Green" title="E40">A22</font></i></a>, <a class="txt" href="#E4:65_2_2"><i><font color="Green" title="E41">A23</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</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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