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<b>let </b><font color="Maroon" title="c1">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">i</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> ;<br/>



<b>assume </b><a NAME="E1:64"/>
<font color="Maroon" title="c2">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>

<a NAME="E2:64"/><b>then </b>
( <font color="Maroon" title="c2">i</font> <a href="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">i</font></span>)</span> <a href="funcop_1.html#K2" title="FUNCOP_1:func.2">--&gt;</a> <font color="Maroon" title="c1">x</font> &amp;  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c2">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a>, <a class="ref" href="finseq_2.html#D2" title="FINSEQ_2:def.2">FINSEQ_2:def 2</a></i>;<br/>
<a NAME="E3:64"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T14" title="FUNCOP_1:th.14">FUNCOP_1:14</a></i>;<br/>
<a NAME="E4:64"/><b>then </b>
( ( <font color="Maroon" title="c1">x</font> is   <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a> implies  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <font color="Maroon" title="c1">x</font></span>)</span> is <a href="trees_3.html#V1" title="TREES_3:attr.1">constituted-Trees</a> ) &amp; (  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <font color="Maroon" title="c1">x</font></span>)</span> is <a href="trees_3.html#V1" title="TREES_3:attr.1">constituted-Trees</a> implies <font color="Maroon" title="c1">x</font> is   <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a> ) &amp; ( <font color="Maroon" title="c2">i</font> <a href="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <font color="Maroon" title="c1">x</font> is <a href="trees_3.html#V4" title="TREES_3:attr.4">Tree-yielding</a> implies  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <font color="Maroon" title="c1">x</font></span>)</span> is <a href="trees_3.html#V1" title="TREES_3:attr.1">constituted-Trees</a> ) &amp; (  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <font color="Maroon" title="c1">x</font></span>)</span> is <a href="trees_3.html#V1" title="TREES_3:attr.1">constituted-Trees</a> implies <font color="Maroon" title="c2">i</font> <a href="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <font color="Maroon" title="c1">x</font> is <a href="trees_3.html#V4" title="TREES_3:attr.4">Tree-yielding</a> ) )
 
<b>by </b><i><a class="ref" href="trees_3.html#D9" target="_self" title="TREES_3:def.9">Def9</a>, <a class="ref" href="trees_3.html#T13" target="_self" title="TREES_3:th.13">Th13</a></i>;<br/>
<b>hence </b><a NAME="E5:64"/>
( <font color="Maroon" title="c2">i</font> <a href="finseq_2.html#K2" title="FINSEQ_2:func.2">|-&gt;</a> <font color="Maroon" title="c1">x</font> is <a href="trees_3.html#V4" title="TREES_3:attr.4">Tree-yielding</a> iff <font color="Maroon" title="c1">x</font> is   <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a> )
 ;<br/>


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