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<span class="kw">let </span><font color="Maroon" title="c1">f</font>, <font color="Maroon" title="c2">g</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="funct_4.html#K2" title="FUNCT_4:func.2">~</a> <span class="p1"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  implies <span class="p1"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:12"/><span class="lab"><font color="Green" title="E10">A1</font></span>: 
 <a href="funct_4.html#K2" title="FUNCT_4:func.2">~</a> <span class="p1"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">x1</font>, <font color="Maroon" title="c4">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">x1</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"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span> or  not <font color="Maroon" title="c4">x2</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"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span> or  not <span class="p1"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x2</font> or <font color="Maroon" title="c3">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x2</font> )</span><br/>

<a NAME="E2:12"/><span class="lab"><font color="Green" title="E11">A2</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">g</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">f</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D8" title="FUNCT_3:def.8">FUNCT_3:def 8</a></span>;<br/>
<a NAME="E3:12"/><span class="lab"><font color="Green" title="E12">A3</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">f</font></span>)</span>,<span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">g</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D8" title="FUNCT_3:def.8">FUNCT_3:def 8</a></span>;<br/>
<span class="kw">assume </span><a NAME="E4:12"/>
<font color="Maroon" title="c3">x1</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"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">x2</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"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span> or  not <span class="p1"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x2</font> or <font color="Maroon" title="c3">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x2</font> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">x11</font>, <font color="Maroon" title="c6">x12</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E6:12"/><span class="lab"><font color="Green" title="E13">A4</font></span>: 
<font color="Maroon" title="c5">x11</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> <font color="Maroon" title="c2">g</font>
 <span class="kw">and </span><br/><a NAME="E7:12"/><span class="lab"><font color="Green" title="E14">A5</font></span>: 
<font color="Maroon" title="c6">x12</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> <font color="Maroon" title="c1">f</font>
 <span class="kw">and </span><br/><a NAME="E8:12"/><span class="lab"><font color="Green" title="E15">A6</font></span>: 
<font color="Maroon" title="c3">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x11</font>,<font color="Maroon" title="c6">x12</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:12"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="ref" href="zfmisc_1.html#T84" title="ZFMISC_1:th.84">ZFMISC_1:84</a></span>;<br/>
<span class="kw">assume </span><a NAME="E9:12"/>
<font color="Maroon" title="c4">x2</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"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <span class="p1"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x2</font> or <font color="Maroon" title="c3">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x2</font> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">x21</font>, <font color="Maroon" title="c8">x22</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E11:12"/><span class="lab"><font color="Green" title="E16">A7</font></span>: 
<font color="Maroon" title="c7">x21</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> <font color="Maroon" title="c2">g</font>
 <span class="kw">and </span><br/><a NAME="E12:12"/><span class="lab"><font color="Green" title="E17">A8</font></span>: 
<font color="Maroon" title="c8">x22</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> <font color="Maroon" title="c1">f</font>
 <span class="kw">and </span><br/><a NAME="E13:12"/><span class="lab"><font color="Green" title="E18">A9</font></span>: 
<font color="Maroon" title="c4">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x21</font>,<font color="Maroon" title="c8">x22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:12"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="ref" href="zfmisc_1.html#T84" title="ZFMISC_1:th.84">ZFMISC_1:84</a></span>;<br/>
<a NAME="E14:12"/>
<font color="Maroon" title="c3">x1</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"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:12"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="txt" href="#E6:12"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="txt" href="#E7:12"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="txt" href="#E8:12"><span class="lab"><font color="Green" title="E15">A6</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/>
<a NAME="E15:12"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A10</font></span>: 
<font color="Maroon" title="c3">x1</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="funct_4.html#K2" title="FUNCT_4:func.2">~</a> <span class="p2"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:12"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="txt" href="#E3:12"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="ref" href="funct_4.html#T46" title="FUNCT_4:th.46">FUNCT_4:46</a></span>;<br/>
<a NAME="E16:12"/>
<font color="Maroon" title="c4">x2</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"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:12"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="txt" href="#E11:12"><span class="lab"><font color="Green" title="E16">A7</font></span></a>, <a class="txt" href="#E12:12"><span class="lab"><font color="Green" title="E17">A8</font></span></a>, <a class="txt" href="#E13:12"><span class="lab"><font color="Green" title="E18">A9</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" title="ZFMISC_1:th.87">ZFMISC_1:87</a></span>;<br/>
<a NAME="E17:12"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A11</font></span>: 
<font color="Maroon" title="c4">x2</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="funct_4.html#K2" title="FUNCT_4:func.2">~</a> <span class="p2"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:12"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="txt" href="#E3:12"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="ref" href="funct_4.html#T46" title="FUNCT_4:th.46">FUNCT_4:46</a></span>;<br/>
<span class="kw">assume </span><a NAME="E18:12"/><span class="lab"><font color="Green" title="E21">A12</font></span>: 
<span class="p1"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x2</font>
 ; <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="c3">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x2</font></span><br/>

<a NAME="E19:12"/><span class="lab"><font color="Green" title="E22">A13</font></span>: 
<span class="p1"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c5">x11</font>,<font color="Maroon" title="c6">x12</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x11</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x12</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:12"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="txt" href="#E7:12"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="ref" href="funct_3.html#D8" title="FUNCT_3:def.8">FUNCT_3:def 8</a></span>;<br/>
<a NAME="E20:12"/><span class="lab"><font color="Green" title="E23">A14</font></span>: 
<span class="p1"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c1">f</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x21</font>,<font color="Maroon" title="c8">x22</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x21</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x22</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:12"><span class="lab"><font color="Green" title="E16">A7</font></span></a>, <a class="txt" href="#E12:12"><span class="lab"><font color="Green" title="E17">A8</font></span></a>, <a class="ref" href="funct_3.html#D8" title="FUNCT_3:def.8">FUNCT_3:def 8</a></span>;<br/>
<a NAME="E21:12"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A15</font></span>: 
<font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x22</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x12</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:12"><span class="lab"><font color="Green" title="E15">A6</font></span></a>, <a class="txt" href="#E13:12"><span class="lab"><font color="Green" title="E18">A9</font></span></a>, <a class="txt" href="#E18:12"><span class="lab"><font color="Green" title="E21">A12</font></span></a>, <a class="txt" href="#E19:12"><span class="lab"><font color="Green" title="E22">A13</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/>
<a NAME="E22:12"/><span class="lab"><font color="Green" title="E25">A16</font></span>: 
<font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x21</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:12"><span class="lab"><font color="Green" title="E15">A6</font></span></a>, <a class="txt" href="#E13:12"><span class="lab"><font color="Green" title="E18">A9</font></span></a>, <a class="txt" href="#E18:12"><span class="lab"><font color="Green" title="E21">A12</font></span></a>, <a class="txt" href="#E19:12"><span class="lab"><font color="Green" title="E22">A13</font></span></a>, <a class="txt" href="#E20:12"><span class="lab"><font color="Green" title="E23">A14</font></span></a>, <a class="ref" href="zfmisc_1.html#T27" title="ZFMISC_1:th.27">ZFMISC_1:27</a></span>;<br/>
<span class="p1">(<span class="default"><a href="funct_4.html#K2" title="FUNCT_4:func.2">~</a> <span class="p2"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">x11</font>,<font color="Maroon" title="c6">x12</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> = 
<span class="p1">(<span class="default"><a href="funct_4.html#K2" title="FUNCT_4:func.2">~</a> <span class="p2"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c5">x11</font>,<font color="Maroon" title="c6">x12</font>)

<br/>.= 
<span class="p1"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c6">x12</font>,<font color="Maroon" title="c5">x11</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:12"><span class="lab"><font color="Green" title="E15">A6</font></span></a>, <a class="txt" href="#E15:12"><span class="lab"><font color="Green" title="E19">A10</font></span></a>, <a class="ref" href="funct_4.html#T43" title="FUNCT_4:th.43">FUNCT_4:43</a></span>
<br/>.= 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">x22</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x21</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:12"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="txt" href="#E7:12"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="txt" href="#E21:12"><span class="lab"><font color="Green" title="E24">A15</font></span></a>, <a class="txt" href="#E22:12"><span class="lab"><font color="Green" title="E25">A16</font></span></a>, <a class="ref" href="funct_3.html#D8" title="FUNCT_3:def.8">FUNCT_3:def 8</a></span>
<br/>.= 
<span class="p1"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c8">x22</font>,<font color="Maroon" title="c7">x21</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:12"><span class="lab"><font color="Green" title="E16">A7</font></span></a>, <a class="txt" href="#E12:12"><span class="lab"><font color="Green" title="E17">A8</font></span></a>, <a class="ref" href="funct_3.html#D8" title="FUNCT_3:def.8">FUNCT_3:def 8</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_4.html#K2" title="FUNCT_4:func.2">~</a> <span class="p2"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x21</font>,<font color="Maroon" title="c8">x22</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:12"><span class="lab"><font color="Green" title="E18">A9</font></span></a>, <a class="txt" href="#E17:12"><span class="lab"><font color="Green" title="E20">A11</font></span></a>, <a class="ref" href="funct_4.html#T43" title="FUNCT_4:th.43">FUNCT_4:43</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_4.html#K2" title="FUNCT_4:func.2">~</a> <span class="p2"><a href="funct_3.html#K15" title="FUNCT_3:func.15">[:</a><span class="default"><font color="Maroon" title="c1">f</font>,<font color="Maroon" title="c2">g</font></span><a href="funct_3.html#K15" title="FUNCT_3:func.15">:]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">x21</font>,<font color="Maroon" title="c8">x22</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>

;<br/>
<span class="kw">hence </span><a NAME="E24:12"/>
<font color="Maroon" title="c3">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="txt" href="#E8:12"><span class="lab"><font color="Green" title="E15">A6</font></span></a>, <a class="txt" href="#E13:12"><span class="lab"><font color="Green" title="E18">A9</font></span></a>, <a class="txt" href="#E15:12"><span class="lab"><font color="Green" title="E19">A10</font></span></a>, <a class="txt" href="#E17:12"><span class="lab"><font color="Green" title="E20">A11</font></span></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</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>
