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<b>let </b><font color="Maroon" title="c1">V</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"> for <font color="Olive" title="b1">f</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="ens_1.html#K12" title="ENS_1:func.12">Ens</a> <font color="Maroon" title="c1">V</font></span>)</span> holds <br/> ( <font color="Olive" title="b1">f</font> is <a href="cat_1.html#V4" title="CAT_1:attr.4">monic</a> iff <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  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="c2">f</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="ens_1.html#K12" title="ENS_1:func.12">Ens</a> <font color="Maroon" title="c1">V</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">f</font> is <a href="cat_1.html#V4" title="CAT_1:attr.4">monic</a> iff <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> )</span><br/>



<b>set </b><font color="Maroon" title="c3">m</font> =  <a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font>;<br/>
<b>thus </b><a NAME="E1:53"/>
( <font color="Maroon" title="c2">f</font> is <a href="cat_1.html#V4" title="CAT_1:attr.4">monic</a> implies <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> implies <font color="Maroon" title="c2">f</font> is <a href="cat_1.html#V4" title="CAT_1:attr.4">monic</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:53_1"/><i><font color="Green" title="E34">A1</font></i>: 
<font color="Maroon" title="c2">f</font> is <a href="cat_1.html#V4" title="CAT_1:attr.4">monic</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a></span><br/>

<b>set </b><font color="Maroon" title="c4">A</font> =  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span>;<br/>
<b>let </b><font color="Maroon" title="c5">x1</font>, <font color="Maroon" title="c6">x2</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"> (  not <font color="Maroon" title="c5">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">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> or  not <font color="Maroon" title="c6">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">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> or  not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x2</font> or <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x2</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E2:53_1"/><i><font color="Green" title="E35">A2</font></i>: 
( <font color="Maroon" title="c5">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"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> &amp; <font color="Maroon" title="c6">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"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> )
 <b>and </b><br/><a NAME="E3:53_1"/><i><font color="Green" title="E36">A3</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x2</font></span><br/>

<a NAME="E4:53_1"/><i><font color="Green" title="E37">A4</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="ens_1.html#E17"><i><font color="Green" title="E10">Lm3</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">A</font> =  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">V</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c8">fx1</font> = <font color="Maroon" title="c7">A</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c5">x1</font>, <font color="Maroon" title="c9">fx2</font> = <font color="Maroon" title="c7">A</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c6">x2</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c7">A</font> <b>by </b><i><a class="txt" href="ens_1.html#E2:53_1"><i><font color="Green" title="E35">A2</font></i></a>, <a class="ref" href="funcop_1.html#T57" title="FUNCOP_1:th.57">FUNCOP_1:57</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">m1</font> = <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c7">A</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Maroon" title="c8">fx1</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>, <font color="Maroon" title="c11">m2</font> = <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">A</font>,<font color="Maroon" title="c7">A</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<font color="Maroon" title="c9">fx2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ens_1.html#K2" title="ENS_1:func.2">Maps</a> <font color="Maroon" title="c1">V</font> <b>by </b><i><a class="ref" href="ens_1.html#T5" target="_self" title="ENS_1:th.5">Th5</a></i>;<br/>
<b>set </b><font color="Maroon" title="c12">f1</font> =  <a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c10">m1</font>;<br/>
<b>set </b><font color="Maroon" title="c13">f2</font> =  <a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c11">m2</font>;<br/>
<a NAME="E8:53_1"/><i><font color="Green" title="E38">A5</font></i>: 
( <font color="Maroon" title="c10">m1</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"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c10">m1</font></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c10">m1</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">m1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c11">m2</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"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c11">m2</font></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c11">m2</font></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">m2</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 
<b>by </b><i><a class="ref" href="ens_1.html#T8" target="_self" title="ENS_1:th.8">Th8</a></i>;<br/>
<a NAME="E9:53_1"/><b>then </b><i><font color="Green" title="E39">A6</font></i>: 
(  <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c10">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A</font> &amp;  <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <font color="Maroon" title="c11">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A</font> &amp;  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c10">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A</font> &amp;  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <font color="Maroon" title="c11">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A</font> )
 
<b>by </b><i><a class="txt" href="ens_1.html#E10"><i><font color="Green" title="E6">Lm1</font></i></a></i>;<br/>
<a NAME="E10:53_1"/>
(  <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p2">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c10">m1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A</font> &amp;  <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p2">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c11">m2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A</font> &amp;  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p2">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c10">m1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A</font> &amp;  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p2">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c11">m2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A</font> )
 
<b>by </b><i><a class="txt" href="ens_1.html#E8:53_1"><i><font color="Green" title="E38">A5</font></i></a>, <a class="txt" href="ens_1.html#E10"><i><font color="Green" title="E6">Lm1</font></i></a></i>;<br/>
<a NAME="E11:53_1"/><b>then </b><i><font color="Green" title="E40">A7</font></i>: 
(  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c10">m1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c11">m2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A</font> &amp;  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c10">m1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A</font> &amp;  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c11">m2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A</font> &amp;  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A</font> )
 
<b>by </b><i><a class="txt" href="ens_1.html#E4:53_1"><i><font color="Green" title="E37">A4</font></i></a>, <a class="ref" href="ens_1.html#D10" target="_self" title="ENS_1:def.10">Def10</a>, <a class="ref" href="ens_1.html#D11" target="_self" title="ENS_1:def.11">Def11</a></i>;<br/>
<b>set </b><font color="Maroon" title="c14">h1</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p3">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c10">m1</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c15">h2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p3">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c11">m2</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c16">ff1</font> = <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p2">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c10">m1</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c17">ff2</font> = <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p2">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c11">m2</font></span>)</span></span>)</span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1"><b>now </b></a><div class="add"><a NAME="E1:53_1_1"/><i><font color="Green" title="E41">A8</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p3">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c10">m1</font></span>)</span></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p3">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c10">m1</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> &amp;  <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p3">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c10">m1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p2">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c10">m1</font></span>)</span></span>)</span> &amp;  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p3">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c10">m1</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p3">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c11">m2</font></span>)</span></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p3">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c11">m2</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> &amp;  <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p3">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c11">m2</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p2">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c11">m2</font></span>)</span></span>)</span> &amp;  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p3">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c11">m2</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> )
 <b>by </b><i><a class="txt" href="ens_1.html#E4:53_1"><i><font color="Green" title="E37">A4</font></i></a>, <a class="txt" href="ens_1.html#E9:53_1"><i><font color="Green" title="E39">A6</font></i></a>, <a class="ref" href="ens_1.html#T12" target="_self" title="ENS_1:th.12">Th12</a></i>;<br/><b>hence </b><a NAME="E2:53_1_1"/>
( <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p2">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c10">m1</font></span>)</span></span>)</span> <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"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <span class="p4">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p5">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c10">m1</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <span class="p4">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p5">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c10">m1</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p2">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c11">m2</font></span>)</span></span>)</span> <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"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <span class="p4">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p5">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c11">m2</font></span>)</span></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <span class="p4">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p5">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c11">m2</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="ens_1.html#T8" target="_self" title="ENS_1:th.8">Th8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p3">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c10">m1</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p3">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c11">m2</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1_1_1"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:53_1_1_1"/><i><font color="Green" title="E42">A9</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p4">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c10">m1</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p4">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c11">m2</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">A</font> )
 <b>by </b><i><a class="txt" href="ens_1.html#E9:53_1"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="ens_1.html#E1:53_1_1"><i><font color="Green" title="E41">A8</font></i></a>, <a class="txt" href="ens_1.html#E17"><i><font color="Green" title="E10">Lm3</font></i></a></i>; <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> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p4">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c10">m1</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p4">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c11">m2</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span><br/><b>let </b><font color="Maroon" title="c18">x</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"> ( <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p4">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c10">m1</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p4">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c11">m2</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> )</span><br/><b>assume </b><a NAME="E2:53_1_1_1"/><i><font color="Green" title="E43">A10</font></i>: 
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p4">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c10">m1</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p4">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c11">m2</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font></span><br/><a NAME="E3:53_1_1_1"/>
( <font color="Maroon" title="c8">fx1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">m1</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  &amp; <font color="Maroon" title="c9">fx2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">m2</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  )
 <b>by </b><i><a class="txt" href="ens_1.html#E8:53_1"><i><font color="Green" title="E38">A5</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><a NAME="E4:53_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p3">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c10">m1</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x1</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p3">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c11">m2</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x2</font> )
 <b>by </b><i><a class="txt" href="ens_1.html#E2:53_1_1_1"><i><font color="Green" title="E43">A10</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/><a NAME="E5:53_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p4">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c10">m1</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x1</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p4">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c11">m2</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x2</font> )
 <b>by </b><i><a class="txt" href="ens_1.html#E1:53_1_1_1"><i><font color="Green" title="E42">A9</font></i></a>, <a class="txt" href="ens_1.html#E2:53_1_1_1"><i><font color="Green" title="E43">A10</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/><b>hence </b><a NAME="E6:53_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p4">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c10">m1</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p4">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c11">m2</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font>
 <b>by </b><i><a class="txt" href="ens_1.html#E3:53_1"><i><font color="Green" title="E36">A3</font></i></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>hence </b><a NAME="E4:53_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p3">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c10">m1</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p3">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c11">m2</font></span>)</span></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</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>
<a NAME="E13:53_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p2">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c10">m1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p2">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c11">m2</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c2">f</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c10">m1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p2">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c10">m1</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c2">f</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <span class="p1">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c11">m2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <span class="p2">(<span class="default"><a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c11">m2</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="ens_1.html#E9:53_1"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="ens_1.html#E11:53_1"><i><font color="Green" title="E40">A7</font></i></a>, <a class="ref" href="ens_1.html#T28" target="_self" title="ENS_1:th.28">Th28</a></i>;<br/>
<a NAME="E14:53_1"/><b>then </b>
(  <a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c10">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c11">m2</font> &amp;  <a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c10">m1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">m1</font> &amp;  <a href="ens_1.html#K15" title="ENS_1:func.15">@</a> <font color="Maroon" title="c11">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">m2</font> )
 
<b>by </b><i><a class="txt" href="ens_1.html#E1:53_1"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="ens_1.html#E11:53_1"><i><font color="Green" title="E40">A7</font></i></a>, <a class="ref" href="cat_1.html#D10" title="CAT_1:def.10">CAT_1:def 10</a></i>;<br/>
<a NAME="E15:53_1"/><b>then </b>
<font color="Maroon" title="c8">fx1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">fx2</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<b>hence </b><font color="Maroon" title="c5">x1</font> = 
<font color="Maroon" title="c9">fx2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x1</font>
<b>by </b><i><a class="txt" href="ens_1.html#E2:53_1"><i><font color="Green" title="E35">A2</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
<br/>.= 
<font color="Maroon" title="c6">x2</font>
<b>by </b><i><a class="txt" href="ens_1.html#E2:53_1"><i><font color="Green" title="E35">A2</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
;<br/> <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="E2:53"/><i><font color="Green" title="E34">A11</font></i>: 
<span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">f</font> is <a href="cat_1.html#V4" title="CAT_1:attr.4">monic</a></span><br/>

<b>let </b><font color="Maroon" title="c4">f1</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="ens_1.html#K12" title="ENS_1:func.12">Ens</a> <font color="Maroon" title="c1">V</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="cat_1.html#D10" title="CAT_1:def.10">CAT_1:def 10</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="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="graph_1.html#U2" title="GRAPH_1:sel.2">Edges</a> of <span class="p1">(<span class="default"><a href="ens_1.html#K12" title="ENS_1:func.12">Ens</a> <font color="Maroon" title="c1">V</font></span>)</span>  holds <br/>(  not  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c4">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Olive">b<sub>1</sub></font> or  not  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c4">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c2">f</font> or  not  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c2">f</font> or  not <font color="Maroon" title="c2">f</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c4">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Olive">b<sub>1</sub></font> or <font color="Maroon" title="c4">f1</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="c5">f2</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="ens_1.html#K12" title="ENS_1:func.12">Ens</a> <font color="Maroon" title="c1">V</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c4">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c5">f2</font> or  not  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c4">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c2">f</font> or  not  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c5">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c2">f</font> or  not <font color="Maroon" title="c2">f</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c4">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c5">f2</font> or <font color="Maroon" title="c4">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f2</font> )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E3:53"/><i><font color="Green" title="E35">A12</font></i>: 
 <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c4">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c5">f2</font>
 <b>and </b><br/><a NAME="E4:53"/><i><font color="Green" title="E36">A13</font></i>: 
(  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c4">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c2">f</font> &amp;  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c5">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c2">f</font> )
 <b>and </b><br/><a NAME="E5:53"/><i><font color="Green" title="E37">A14</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c4">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c5">f2</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">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f2</font></span><br/>

<b>set </b><font color="Maroon" title="c6">m1</font> =  <a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c4">f1</font>;<br/>
<b>set </b><font color="Maroon" title="c7">m2</font> =  <a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c5">f2</font>;<br/>
<a NAME="E6:53"/><i><font color="Green" title="E38">A15</font></i>: 
(  <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c4">f1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c4">f1</font> &amp;  <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c5">f2</font> &amp;  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c4">f1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c4">f1</font> &amp;  <a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c5">f2</font> &amp;  <a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c2">f</font> )
 
<b>by </b><i><a class="ref" href="ens_1.html#D10" target="_self" title="ENS_1:def.10">Def10</a>, <a class="ref" href="ens_1.html#D11" target="_self" title="ENS_1:def.11">Def11</a></i>;<br/>
<a NAME="E7:53"/><b>then </b>
( <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c4">f1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c4">f1</font> &amp; <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="cat_1.html#K5" title="CAT_1:func.5">*</a> <font color="Maroon" title="c5">f2</font> &amp; <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c4">f1</font></span>)</span> <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"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <span class="p4">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c4">f1</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <span class="p4">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c4">f1</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="ens_1.html#K6" title="ENS_1:func.6">*</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c5">f2</font></span>)</span> <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"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <span class="p4">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c5">f2</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <span class="p4">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 
<b>by </b><i><a class="txt" href="ens_1.html#E4:53"><i><font color="Green" title="E36">A13</font></i></a>, <a class="ref" href="ens_1.html#D7" target="_self" title="ENS_1:def.7">Def7</a>, <a class="ref" href="ens_1.html#T28" target="_self" title="ENS_1:th.28">Th28</a></i>;<br/>
<a NAME="E8:53"/><b>then </b>
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c4">f1</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c4">f1</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c5">f2</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_1.html#K3" title="CAT_1:func.3">dom</a> <font color="Maroon" title="c2">f</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c4">f1</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c4">f1</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cat_1.html#K4" title="CAT_1:func.4">cod</a> <font color="Maroon" title="c5">f2</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c4">f1</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span> )
 
<b>by </b><i><a class="txt" href="ens_1.html#E5:53"><i><font color="Green" title="E37">A14</font></i></a>, <a class="txt" href="ens_1.html#E6:53"><i><font color="Green" title="E38">A15</font></i></a>, <a class="txt" href="ens_1.html#E17"><i><font color="Green" title="E10">Lm3</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/>
<a NAME="E9:53"/><b>then </b>
( <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c4">f1</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  &amp;  <a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c4">f1</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"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <span class="p4">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c4">f1</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <span class="p4">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c4">f1</font></span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c4">f1</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp;  <a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c5">f2</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"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K3" title="ENS_1:func.3">dom</a> <span class="p4">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c5">f2</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="ens_1.html#K4" title="ENS_1:func.4">cod</a> <span class="p4">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c5">f2</font></span>)</span></span>)</span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ens_1.html#K16" title="ENS_1:func.16">@</a> <font color="Maroon" title="c5">f2</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 
<b>by </b><i><a class="txt" href="ens_1.html#E2:53"><i><font color="Green" title="E34">A11</font></i></a>, <a class="txt" href="ens_1.html#E3:53"><i><font color="Green" title="E35">A12</font></i></a>, <a class="txt" href="ens_1.html#E4:53"><i><font color="Green" title="E36">A13</font></i></a>, <a class="ref" href="ens_1.html#T8" target="_self" title="ENS_1:th.8">Th8</a>, <a class="ref" href="funct_1.html#T49" title="FUNCT_1:th.49">FUNCT_1:49</a></i>;<br/>
<b>hence </b><a NAME="E10:53"/>
<font color="Maroon" title="c4">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f2</font>
 <b>by </b><i><a class="txt" href="ens_1.html#E3:53"><i><font color="Green" title="E35">A12</font></i></a>, <a class="txt" href="ens_1.html#E4:53"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="ens_1.html#E6:53"><i><font color="Green" title="E38">A15</font></i></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>
