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<span class="kw">set </span><font color="Maroon" title="c1">X</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">T</font> where <font color="Olive" title="b1">T</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span> : <font color="Olive" title="b1">T</font> is   <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a> </span> } </span> ;<br/>
<span class="kw">take </span>
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">T</font> where <font color="Olive" title="b1">T</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span> : <font color="Olive" title="b1">T</font> is   <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a> </span> } </span> 
; <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">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b2">T</font> where <font color="Olive" title="b2">T</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span> : <font color="Olive" title="b2">T</font> is   <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a> </span> } </span>  iff <font color="Olive" title="b1">x</font> is   <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">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"> ( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">T</font> where <font color="Olive" title="b1">T</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span> : <font color="Olive" title="b1">T</font> is   <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a> </span> } </span>  iff <font color="Maroon" title="c2">x</font> is   <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a> )</span><br/>

<span class="kw">thus </span><a NAME="E1:2_1_1"/>
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">T</font> where <font color="Olive" title="b1">T</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span> : <font color="Olive" title="b1">T</font> is   <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a> </span> } </span>  implies <font color="Maroon" title="c2">x</font> is   <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"> ( <font color="Maroon" title="c2">x</font> is   <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a> implies <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">T</font> where <font color="Olive" title="b1">T</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span> : <font color="Olive" title="b1">T</font> is   <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a> </span> } </span>  )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="2_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:2_1_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">T</font> where <font color="Olive" title="b1">T</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span> : <font color="Olive" title="b1">T</font> is   <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a> </span> } </span> 
 ; <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">x</font> is   <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a></span><br/>

<a NAME="E2:2_1_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">T</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span> st <br/>( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">T</font> &amp; <font color="Olive" title="b1">T</font> is   <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:2_1_1_1"/>
<font color="Maroon" title="c2">x</font> is   <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/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:2_1_1"/>
<font color="Maroon" title="c2">x</font> is   <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"> <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">T</font> where <font color="Olive" title="b1">T</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span> : <font color="Olive" title="b1">T</font> is   <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a> </span> } </span> </span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">T</font> = <font color="Maroon" title="c2">x</font> as   <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a> ;<br/>
<a NAME="E4:2_1_1"/>
<font color="Maroon" title="c3">T</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="trees_1.html#D3" onmouseover="rs('trees_1/D3')" onmouseout="rh()">TREES_1:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:2_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">T</font> where <font color="Olive" title="b1">T</font> is   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span> : <font color="Olive" title="b1">T</font> is   <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a> </span> } </span> 
 ; <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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