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<span class="kw">let </span><font color="Maroon" title="c1">k</font>, <font color="Maroon" title="c2">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">m</font> iff <font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )</span><br/>

<div><span class="lab"><font color="Green" title="E156">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="183_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:183_1"/>
<font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">m</font></span><br/><a NAME="E2:183_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E157">A2</font></span>: 
<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></span>;<br/><a NAME="E3:183_1"/>
<font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></span>;<br/><span class="kw">hence </span><a NAME="E4:183_1"/>
<font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:183_1"><span class="lab"><font color="Green" title="E157">A2</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="183_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:183_2"/>
<font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span><br/><a NAME="E2:183_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T20" title="INT_1:th.20">INT_1:20</a></span>;<br/><span class="kw">hence </span><a NAME="E3:183_2"/>
<font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T21" title="XREAL_1:th.21">XREAL_1:21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:183"/>
( <font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">m</font> iff <font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:183"><span class="lab"><font color="Green" title="E156">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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