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<b>let </b><font color="Maroon">i1</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">i2</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<b>assume </b><a NAME="E1:43"/>
( <font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> or <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> )
 ;<br/>

<a NAME="E2:43"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font>
 
;<br/>
<b>hence </b><a NAME="E3:43"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="binarith.html#K3" target="_self">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="binarith.html#K3" target="_self">-'</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> )
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>

<a NAME="E4:43"/>
<font color="Maroon">i1</font> <a href="binarith.html#K3" target="_self">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:43"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3" target="_self">-'</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font> <a href="xcmplx_0.html#K2">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E6:43"/><b>then </b><i><font color="Green">E40</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3" target="_self">-'</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i2</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E7:43"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3" target="_self">-'</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K3" target="_self">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3" target="_self">-'</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:43"/>
( <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3" target="_self">-'</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i2</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3" target="_self">-'</a> 1</span>)</span> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="binarith.html#K3" target="_self">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i2</font> )
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>

<b>thus </b><a NAME="E9:43"/>
( <font color="Maroon">i1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">i1</font> <a href="binarith.html#K3" target="_self">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">i1</font> <a href="binarith.html#K3" target="_self">-'</a> 2 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">i1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> )
 <b>by </b><i>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>


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