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<b>let </b><font color="Maroon">it1</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="dtconstr.html#K4" target="_self">TS</a> <a href="dtconstr.html#K5" target="_self">PeanoNat</a> , <font color="Maroon">it2</font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="dtconstr.html#K4" target="_self">TS</a> <a href="dtconstr.html#K5" target="_self">PeanoNat</a> ;<br/>

<b>assume </b><a NAME="E1:48_1_2"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">it1</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K2">root-tree</a> 0 &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">it1</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="dtconstr.html#K13" target="_self">PNsucc</a> <span class="p1">(<span class="default"><font color="Maroon">it1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span> ) &amp; <font color="Maroon">it2</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K2">root-tree</a> 0 &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">it2</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="dtconstr.html#K13" target="_self">PNsucc</a> <span class="p1">(<span class="default"><font color="Maroon">it2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span> ) &amp; not <font color="Maroon">it1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">it2</font> )
 ;<br/>

<a NAME="E2:48_1_2"/><b>then </b><i><font color="Green">E32</font></i>: 
( <font color="Maroon">it1</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="dtconstr.html#K8" target="_self">root-tree</a> <font color="Maroon">O</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">it1</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">n</font>,<font color="Maroon">it1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font>) ) )
 
;<br/>
<a NAME="E3:48_1_2"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">it2</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="dtconstr.html#K8" target="_self">root-tree</a> <font color="Maroon">O</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">it2</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">n</font>,<font color="Maroon">it2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font>) ) )
 
<b>by </b><i><a class="txt" href="dtconstr.html#E2:48_1_2"><i><font color="Green">E32</font></i></a></i>;<br/>
<a NAME="E4:48_1_2"/>
<font color="Maroon">it1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">it2</font>
 
<b>from </b><i><a class="ref" href="recdef_1.html#S12">RECDEF_1:sch 12</a>(, <a class="txt" href="dtconstr.html#E3:48_1_2"><i><font color="Green">E34</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:48_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="dtconstr.html#E2:48_1_2"><i><font color="Green">E32</font></i></a></i>;<br/>


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