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<b>let </b><font color="Maroon" title="c1">H</font> be   <a href="zf_lang.html#NM2" title="ZF_LANG:NM.2">ZF-formula</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a> holds <br/> ( <font color="Maroon" title="c1">H</font> is <a href="zf_lang.html#V2" title="ZF_LANG:attr.2">being_equality</a> iff <font color="Maroon" title="c1">H</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> is <a href="zf_lang.html#V2" title="ZF_LANG:attr.2">being_equality</a> )</span><br/><b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font> be   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">H</font> is <a href="zf_lang.html#V2" title="ZF_LANG:attr.2">being_equality</a> iff <font color="Maroon" title="c1">H</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> is <a href="zf_lang.html#V2" title="ZF_LANG:attr.2">being_equality</a> )</span><br/>





<b>thus </b><a NAME="E1:164"/>
( <font color="Maroon" title="c1">H</font> is <a href="zf_lang.html#V2" title="ZF_LANG:attr.2">being_equality</a> implies <font color="Maroon" title="c1">H</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> is <a href="zf_lang.html#V2" title="ZF_LANG:attr.2">being_equality</a> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">H</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> is <a href="zf_lang.html#V2" title="ZF_LANG:attr.2">being_equality</a> implies <font color="Maroon" title="c1">H</font> is <a href="zf_lang.html#V2" title="ZF_LANG:attr.2">being_equality</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="164_1"><b>proof </b></a><div class="add">

<b>given </b><font color="Maroon" title="c4">x1</font>, <font color="Maroon" title="c5">x2</font> being   <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a><b> such that </b><a NAME="E2:164_1"/><i><font color="Green" title="E67">A1</font></i>: 
<font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Maroon" title="c5">x2</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="zf_lang.html#D10" title="ZF_LANG:def.10">ZF_LANG:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">H</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> is <a href="zf_lang.html#V2" title="ZF_LANG:attr.2">being_equality</a></span><br/>

<a NAME="E3:164_1"/>
 ex <font color="Olive" title="b1">z1</font>, <font color="Olive" title="b2">z2</font> being  <a href="zf_lang.html#NM1" title="ZF_LANG:NM.1">Variable</a> st <font color="Maroon" title="c1">H</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">z1</font> <a href="zf_lang.html#K4" title="ZF_LANG:func.4">'='</a> <font color="Olive" title="b2">z2</font>
 
<b>by </b><i><a class="txt" href="#E2:164_1"><i><font color="Green" title="E67">A1</font></i></a>, <a class="ref" href="zf_lang1.html#T167" target="_self" title="ZF_LANG1:th.167">Th167</a></i>;<br/>
<b>hence </b><a NAME="E4:164_1"/>
<font color="Maroon" title="c1">H</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> is <a href="zf_lang.html#V2" title="ZF_LANG:attr.2">being_equality</a>
 <b>by </b><i><a class="ref" href="zf_lang.html#T16" title="ZF_LANG:th.16">ZF_LANG:16</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>assume </b><a NAME="E2:164"/><i><font color="Green" title="E67">A2</font></i>: 
<font color="Maroon" title="c1">H</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> is <a href="zf_lang.html#V2" title="ZF_LANG:attr.2">being_equality</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">H</font> is <a href="zf_lang.html#V2" title="ZF_LANG:attr.2">being_equality</a></span><br/>

<a NAME="E3:164"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3 &amp; 3 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">H</font> )
 
<b>by </b><i><a class="ref" href="zf_lang.html#T29" title="ZF_LANG:th.29">ZF_LANG:29</a></i>;<br/>
<a NAME="E4:164"/><b>then </b>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">H</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">H</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E5:164"/><b>then </b><i><font color="Green" title="E68">A3</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c1">H</font> <a href="zf_lang1.html#K6" title="ZF_LANG1:func.6">/</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <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="c3">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">H</font> )
 
<b>by </b><i><a class="txt" href="#E2:164"><i><font color="Green" title="E67">A2</font></i></a>, <a class="ref" href="zf_lang1.html#T148" target="_self" title="ZF_LANG1:th.148">Th148</a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a>, <a class="ref" href="zf_lang.html#T35" title="ZF_LANG:th.35">ZF_LANG:35</a></i>;<br/>
<a NAME="E6:164"/><b>then </b>
<font color="Maroon" title="c1">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">x</font>
 
<b>by </b><i><a class="ref" href="zf_lang1.html#D4" target="_self" title="ZF_LANG1:def.4">Def4</a></i>;<br/>
<a NAME="E7:164"/><b>then </b>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">H</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 
<b>by </b><i><a class="txt" href="#E5:164"><i><font color="Green" title="E68">A3</font></i></a>, <a class="ref" href="zf_lang1.html#D4" target="_self" title="ZF_LANG1:def.4">Def4</a></i>;<br/>
<b>hence </b><a NAME="E8:164"/>
<font color="Maroon" title="c1">H</font> is <a href="zf_lang.html#V2" title="ZF_LANG:attr.2">being_equality</a>
 <b>by </b><i><a class="ref" href="zf_lang.html#T41" title="ZF_LANG:th.41">ZF_LANG:41</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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