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<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>





<span class="kw">thus </span><a NAME="E1:163"/>
( <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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="163_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">given </span><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><span class="kw"> such that </span><a NAME="E2:163_1"/><span class="lab"><font color="Green" title="E66">A1</font></span>: 
<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>
 ; <span class="comment"><font color="firebrick">:: according to </font></span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="E3:163_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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:163_1"><span class="lab"><font color="Green" title="E66">A1</font></span></a>, <a class="ref" href="zf_lang1.html#T153" target="_self" title="ZF_LANG1:th.153">Th167</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:163_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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#T5" title="ZF_LANG:th.5">ZF_LANG:5</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/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:163"/>
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="E3:163"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A2</font></span>: 
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#T18" title="ZF_LANG:th.18">ZF_LANG:18</a></span>;<br/>
<a NAME="E4:163"/>
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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#T13" title="ZF_LANG:th.13">ZF_LANG:13</a></span>;<br/>
<a NAME="E5:163"/><span class="kw">then </span>
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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E6:163"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A3</font></span>: 
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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E7:163"/>
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang1.html#T135" target="_self" title="ZF_LANG1:th.135">Th148</a></span>;<br/>
<a NAME="E8:163"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:163"><span class="lab"><font color="Green" title="E66">A2</font></span></a>, <a class="txt" href="#E6:163"><span class="lab"><font color="Green" title="E67">A3</font></span></a>, <a class="ref" href="zf_lang1.html#D3" target="_self" title="ZF_LANG1:def.3">Def4</a></span>;<br/>
<a NAME="E9:163"/><span class="kw">then </span>
 <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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:163"><span class="lab"><font color="Green" title="E66">A2</font></span></a>, <a class="txt" href="#E6:163"><span class="lab"><font color="Green" title="E67">A3</font></span></a>, <a class="ref" href="zf_lang1.html#D3" target="_self" title="ZF_LANG1:def.3">Def4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:163"/>
<font color="Maroon" title="c1">H</font> is  <a href="zf_lang.html#V2" title="ZF_LANG:attr.2">being_equality</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zf_lang.html#T24" title="ZF_LANG:th.24">ZF_LANG:24</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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