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<span class="kw">let </span><font color="Maroon" title="c1">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  holds <br/>( <font color="Olive" title="b1">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#V5" title="TREES_3:attr.5">FinTree-yielding</a>  iff <font color="Maroon" title="c1">x</font> is   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">i</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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#V5" title="TREES_3:attr.5">FinTree-yielding</a>  iff <font color="Maroon" title="c1">x</font> is   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a> ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:61"/><span class="lab"><font color="Green" title="E34">A1</font></span>: 
<font color="Maroon" title="c2">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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#V5" title="TREES_3:attr.5">FinTree-yielding</a>  iff <font color="Maroon" title="c1">x</font> is   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a> )</span><br/>

<a NAME="E2:61"/>
<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#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c1">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#D2" onmouseover="rs('finseq_2/D2')" onmouseout="rh()">FINSEQ_2:def 2</a></span>;<br/>
<a NAME="E3:61"/><span class="kw">then </span>
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:61"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="ref" href="funcop_1.html#T8" onmouseover="rs('funcop_1/T8')" onmouseout="rh()">FUNCOP_1:8</a></span>;<br/>
<a NAME="E4:61"/><span class="kw">then </span>
( <font color="Maroon" title="c1">x</font> is   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a> iff  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">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#V2" title="TREES_3:attr.2">constituted-FinTrees</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="trees_3.html#T13" target="_self" onmouseover="rs('trees_3/T13')" onmouseout="rh()">Th13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:61"/>
( <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#V5" title="TREES_3:attr.5">FinTree-yielding</a>  iff <font color="Maroon" title="c1">x</font> is   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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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