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<b>let </b><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">B</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="pralg_1.html#K6" title="PRALG_1:func.6">Inv</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a></span><br/><b>let </b><font color="Maroon" title="c3">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K6" title="PRALG_1:func.6">Inv</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> or  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K6" title="PRALG_1:func.6">Inv</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> or  not <span class="p1">(<span class="default"><a href="pralg_1.html#K6" title="PRALG_1:func.6">Inv</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K6" title="PRALG_1:func.6">Inv</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> or <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> )</span><br/><b>let </b><font color="Maroon" title="c4">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K6" title="PRALG_1:func.6">Inv</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> or  not <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K6" title="PRALG_1:func.6">Inv</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> or  not <span class="p1">(<span class="default"><a href="pralg_1.html#K6" title="PRALG_1:func.6">Inv</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K6" title="PRALG_1:func.6">Inv</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">y</font> or <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> )</span><br/>





<b>assume </b><a NAME="E1:9"/><i><font color="Green" title="E9">A1</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K6" title="PRALG_1:func.6">Inv</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> &amp; <font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K6" title="PRALG_1:func.6">Inv</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pralg_1.html#K6" title="PRALG_1:func.6">Inv</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K6" title="PRALG_1:func.6">Inv</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">y</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font></span><br/>

<b>then reconsider </b><font color="Maroon" title="c5">x1</font> = <font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c6">y1</font> = <font color="Maroon" title="c4">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:9"/>
( <span class="p1">(<span class="default"><a href="pralg_1.html#K6" title="PRALG_1:func.6">Inv</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">x1</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">x1</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <span class="p1">(<span class="default"><a href="pralg_1.html#K6" title="PRALG_1:func.6">Inv</a> <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">B</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">y1</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">y1</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> </span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> )
 
<b>by </b><i><a class="ref" href="pralg_1.html#D6" target="_self" title="PRALG_1:def.6">Def6</a></i>;<br/>
<a NAME="E4:9"/><b>then </b>
( <font color="Maroon" title="c5">x1</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y1</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a>  &amp; <font color="Maroon" title="c5">x1</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y1</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  )
 
<b>by </b><i><a class="txt" href="pralg_1.html#E1:9"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><a NAME="E5:9"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font>
 <b>by </b><i><a class="ref" href="domain_1.html#T12" title="DOMAIN_1:th.12">DOMAIN_1:12</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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