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<span class="kw">let </span><font color="Maroon" title="c1">f</font> be   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">k1</font>, <font color="Olive" title="b2">k2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">k2</font> holds <br/> <a href="finseq_8.html#K2" title="FINSEQ_8:func.2">smid</a> (<font color="Maroon" title="c1">f</font>,<font color="Olive" title="b1">k1</font>,<font color="Olive" title="b2">k2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_6.html#K2" title="FINSEQ_6:func.2">mid</a> (<font color="Maroon" title="c1">f</font>,<font color="Olive" title="b1">k1</font>,<font color="Olive" title="b2">k2</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c2">k1</font>, <font color="Maroon" title="c3">k2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">k1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k2</font> implies  <a href="finseq_8.html#K2" title="FINSEQ_8:func.2">smid</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">k1</font>,<font color="Maroon" title="c3">k2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_6.html#K2" title="FINSEQ_6:func.2">mid</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">k1</font>,<font color="Maroon" title="c3">k2</font>) )</span><br/>



<span class="kw">assume </span><a NAME="E1:7"/><span class="lab"><font color="Green" title="E3">A1</font></span>: 
<font color="Maroon" title="c2">k1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="finseq_8.html#K2" title="FINSEQ_8:func.2">smid</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">k1</font>,<font color="Maroon" title="c3">k2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_6.html#K2" title="FINSEQ_6:func.2">mid</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">k1</font>,<font color="Maroon" title="c3">k2</font>)</span><br/>

<span class="kw">then </span><span class="p1">(<span class="default"><font color="Maroon" title="c3">k2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c2">k1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">k2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">k1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">k2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c2">k1</font>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">k2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <font color="Maroon" title="c2">k1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E3:7"/>
 <a href="finseq_8.html#K2" title="FINSEQ_8:func.2">smid</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">k1</font>,<font color="Maroon" title="c3">k2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_6.html#K2" title="FINSEQ_6:func.2">mid</a> (<font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">k1</font>,<font color="Maroon" title="c3">k2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="ref" href="finseq_6.html#D3" title="FINSEQ_6:def.3">FINSEQ_6:def 3</a></span>; <a class="txt" 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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