<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c4">IT1</font>, <font color="Maroon" title="c5">IT2</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( not <font color="Maroon" title="c3">W</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> &amp; <font color="Maroon" title="c4">IT1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="glib_005.html#K8" title="GLIB_005:func.8">.flowSeq</a> <font color="Maroon" title="c2">EL</font></span>)</span> &amp; ( for <font color="Olive" title="b1">k</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="glib_005.html#K8" title="GLIB_005:func.8">.flowSeq</a> <font color="Maroon" title="c2">EL</font></span>)</span> holds <br/><font color="Maroon" title="c4">IT1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> ) &amp; <font color="Maroon" title="c5">IT2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="glib_005.html#K8" title="GLIB_005:func.8">.flowSeq</a> <font color="Maroon" title="c2">EL</font></span>)</span> &amp; ( for <font color="Olive" title="b1">k</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="glib_005.html#K8" title="GLIB_005:func.8">.flowSeq</a> <font color="Maroon" title="c2">EL</font></span>)</span> holds <br/><font color="Maroon" title="c5">IT2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> ) implies <font color="Maroon" title="c4">IT1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">IT2</font> ) &amp; ( <font color="Maroon" title="c3">W</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> &amp; <font color="Maroon" title="c4">IT1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c5">IT2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c4">IT1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">IT2</font> ) )</span><br/>

<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">W</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> &amp; <font color="Maroon" title="c4">IT1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c5">IT2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c4">IT1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">IT2</font> )</span>
<div class="add"><b>assume </b><a NAME="E1:35_1_2_1"/>
not <font color="Maroon" title="c3">W</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">IT1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="glib_005.html#K8" title="GLIB_005:func.8">.flowSeq</a> <font color="Maroon" title="c2">EL</font></span>)</span> &amp; ( for <font color="Olive" title="b1">k</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="glib_005.html#K8" title="GLIB_005:func.8">.flowSeq</a> <font color="Maroon" title="c2">EL</font></span>)</span> holds <br/><font color="Maroon" title="c4">IT1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> ) &amp; <font color="Maroon" title="c5">IT2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="glib_005.html#K8" title="GLIB_005:func.8">.flowSeq</a> <font color="Maroon" title="c2">EL</font></span>)</span> &amp; ( for <font color="Olive" title="b1">k</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="glib_005.html#K8" title="GLIB_005:func.8">.flowSeq</a> <font color="Maroon" title="c2">EL</font></span>)</span> holds <br/><font color="Maroon" title="c5">IT2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> ) implies <font color="Maroon" title="c4">IT1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">IT2</font> )</span><br/><b>assume </b><a NAME="E2:35_1_2_1"/><i><font color="Green" title="E30">A4</font></i>: 
( <font color="Maroon" title="c4">IT1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="glib_005.html#K8" title="GLIB_005:func.8">.flowSeq</a> <font color="Maroon" title="c2">EL</font></span>)</span> &amp; ( for <font color="Olive" title="b1">k</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="glib_005.html#K8" title="GLIB_005:func.8">.flowSeq</a> <font color="Maroon" title="c2">EL</font></span>)</span> holds <br/><font color="Maroon" title="c4">IT1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">IT2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="glib_005.html#K8" title="GLIB_005:func.8">.flowSeq</a> <font color="Maroon" title="c2">EL</font></span>)</span> &amp; ( for <font color="Olive" title="b1">k</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="glib_005.html#K8" title="GLIB_005:func.8">.flowSeq</a> <font color="Maroon" title="c2">EL</font></span>)</span> holds <br/><font color="Maroon" title="c5">IT2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> ) implies <font color="Maroon" title="c4">IT1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">IT2</font> )</span><br/><b>assume </b><a NAME="E3:35_1_2_1"/>
( <font color="Maroon" title="c5">IT2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="glib_005.html#K8" title="GLIB_005:func.8">.flowSeq</a> <font color="Maroon" title="c2">EL</font></span>)</span> &amp; ( for <font color="Olive" title="b1">k</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">W</font> <a href="glib_005.html#K8" title="GLIB_005:func.8">.flowSeq</a> <font color="Maroon" title="c2">EL</font></span>)</span> holds <br/><font color="Maroon" title="c5">IT2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">IT1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">IT2</font></span><br/><a NAME="E4:35_1_2_1"/><b>then </b>
( <font color="Maroon" title="c4">IT1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">IT2</font> &amp; <font color="Maroon" title="c5">IT2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">IT1</font> )
 <b>by </b><i><a class="txt" href="#E2:35_1_2_1"><i><font color="Green" title="E30">A4</font></i></a></i>;<br/><b>hence </b><a NAME="E5:35_1_2_1"/>
<font color="Maroon" title="c4">IT1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">IT2</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E2:35_1_2"/>
( <font color="Maroon" title="c3">W</font> is <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a> &amp; <font color="Maroon" title="c4">IT1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c5">IT2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c4">IT1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">IT2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
