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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">with_non-empty_elements</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">o1</font>, <font color="Olive" title="b2">o2</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">X</font></span>)</span>  st <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b1">o1</font>,<font color="Olive" title="b2">o2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/>for <font color="Olive" title="b3">A</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b1">o1</font>,<font color="Olive" title="b2">o2</font><br/> for <font color="Olive" title="b4">F</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b1">o1</font>,<font color="Olive" title="b2">o2</font>  st <font color="Olive" title="b4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">A</font> holds <br/>( <font color="Olive" title="b3">A</font> is  <a href="altcat_3.html#V5" title="ALTCAT_3:attr.5">epi</a>  iff <font color="Olive" title="b4">F</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">o1</font>, <font color="Maroon" title="c3">o2</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">X</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">o1</font>,<font color="Maroon" title="c3">o2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies for <font color="Olive" title="b1">A</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">o1</font>,<font color="Maroon" title="c3">o2</font><br/> for <font color="Olive" title="b2">F</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">o1</font>,<font color="Maroon" title="c3">o2</font>  st <font color="Olive" title="b2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> holds <br/>( <font color="Olive" title="b1">A</font> is  <a href="altcat_3.html#V5" title="ALTCAT_3:attr.5">epi</a>  iff <font color="Olive" title="b2">F</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:24"/><span class="lab"><font color="Green" title="E19">A1</font></span>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">o1</font>,<font color="Maroon" title="c3">o2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">A</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">o1</font>,<font color="Maroon" title="c3">o2</font><br/> for <font color="Olive" title="b2">F</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">o1</font>,<font color="Maroon" title="c3">o2</font>  st <font color="Olive" title="b2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">A</font> holds <br/>( <font color="Olive" title="b1">A</font> is  <a href="altcat_3.html#V5" title="ALTCAT_3:attr.5">epi</a>  iff <font color="Olive" title="b2">F</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">A</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">o1</font>,<font color="Maroon" title="c3">o2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">F</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">o1</font>,<font color="Maroon" title="c3">o2</font>  st <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A</font> holds <br/>( <font color="Maroon" title="c4">A</font> is  <a href="altcat_3.html#V5" title="ALTCAT_3:attr.5">epi</a>  iff <font color="Olive" title="b1">F</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">F</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">o1</font>,<font color="Maroon" title="c3">o2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A</font> implies ( <font color="Maroon" title="c4">A</font> is  <a href="altcat_3.html#V5" title="ALTCAT_3:attr.5">epi</a>  iff <font color="Maroon" title="c5">F</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  ) )</span><br/>



<span class="kw">assume </span><a NAME="E2:24"/><span class="lab"><font color="Green" title="E20">A2</font></span>: 
<font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">A</font> is  <a href="altcat_3.html#V5" title="ALTCAT_3:attr.5">epi</a>  iff <font color="Maroon" title="c5">F</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:24_1"/>
( 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="c1">X</font> holds <br/><font color="Olive" title="b1">x</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  or  ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; not <font color="Olive" title="b1">x</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1"><span class="kw">suppose </span></a><a NAME="E1:24_1_1"/><span class="lab"><font color="Green" title="E21">A3</font></span>: 
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="c1">X</font> holds <br/><font color="Olive" title="b1">x</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">A</font> is  <a href="altcat_3.html#V5" title="ALTCAT_3:attr.5">epi</a>  iff <font color="Maroon" title="c5">F</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  )</span><br/><div class="add"><span class="kw">thus </span><a NAME="E2:24_1_1"/>
( <font color="Maroon" title="c4">A</font> is  <a href="altcat_3.html#V5" title="ALTCAT_3:attr.5">epi</a>  implies <font color="Maroon" title="c5">F</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">F</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  implies <font color="Maroon" title="c4">A</font> is  <a href="altcat_3.html#V5" title="ALTCAT_3:attr.5">epi</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:24_1_1_1"/>
<font color="Maroon" title="c4">A</font> is  <a href="altcat_3.html#V5" title="ALTCAT_3:attr.5">epi</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">F</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> </span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:24_1_1_1_1_1"/>
( <font color="Maroon" title="c3">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c3">o2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:24_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E22">A4</font></span>: 
<font color="Maroon" title="c3">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">F</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> </span><br/><div class="add"><a NAME="E2:24_1_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><span class="kw">hence </span><a NAME="E3:24_1_1_1_1_1_1"/>
<font color="Maroon" title="c5">F</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a>, <a class="ref" href="relat_1.html#T60" title="RELAT_1:th.60">RELAT_1:60</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:24_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E22">A5</font></span>: 
<font color="Maroon" title="c3">o2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">F</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> </span><br/><div class="add"><a NAME="E2:24_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E23">A6</font></span>: 
<font color="Maroon" title="c2">o1</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="altcat_1.html#D16" title="ALTCAT_1:def.16">ALTCAT_1:def 16</a></span>;<br/><a NAME="E3:24_1_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">o1</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1_1"><span class="lab"><font color="Green" title="E21">A3</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:24_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E24">A7</font></span>: 
<font color="Maroon" title="c2">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">z</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E23">A6</font></span></a>, <a class="ref" href="realset1.html#D4" title="REALSET1:def.4">REALSET1:def 4</a></span>;<br/><a NAME="E6:24_1_1_1_1_1_2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">z</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="txt" href="#E5:24_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E24">A7</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><a NAME="E7:24_1_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A8</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">F</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T65" title="RELAT_1:th.65">RELAT_1:65</a></span>;<br/><a NAME="E8:24_1_1_1_1_1_2"/>
<font color="Maroon" title="c3">o2</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="altcat_1.html#D16" title="ALTCAT_1:def.16">ALTCAT_1:def 16</a></span>;<br/><a NAME="E9:24_1_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">o2</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1_1"><span class="lab"><font color="Green" title="E21">A3</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E11:24_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E26">A9</font></span>: 
<font color="Maroon" title="c3">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c7">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="ref" href="realset1.html#D4" title="REALSET1:def.4">REALSET1:def 4</a></span>;<br/><a NAME="E12:24_1_1_1_1_1_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c7">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:24_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E26">A9</font></span></a>, <a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/><a NAME="E13:24_1_1_1_1_1_2"/><span class="kw">then </span>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c7">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:24_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E25">A8</font></span></a>, <a class="ref" href="zfmisc_1.html#T39" title="ZFMISC_1:th.39">ZFMISC_1:39</a></span>;<br/><span class="kw">hence </span><a NAME="E14:24_1_1_1_1_1_2"/>
<font color="Maroon" title="c5">F</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:24_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E26">A9</font></span></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:24_1_1_1"/>
<font color="Maroon" title="c5">F</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">thus </span><a NAME="E3:24_1_1"/>
( <font color="Maroon" title="c5">F</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  implies <font color="Maroon" title="c4">A</font> is  <a href="altcat_3.html#V5" title="ALTCAT_3:attr.5">epi</a>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:24_1_1_2"/><span class="lab"><font color="Green" title="E22">A10</font></span>: 
<font color="Maroon" title="c5">F</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">A</font> is  <a href="altcat_3.html#V5" title="ALTCAT_3:attr.5">epi</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c6">o</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">X</font></span>)</span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="altcat_3.html#D8" target="_self" title="ALTCAT_3:def.8">ALTCAT_3:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">o2</font>,<font color="Maroon" title="c6">o</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies for <font color="Olive" title="b1">B</font>, <font color="Olive" title="b2">C</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">o2</font>,<font color="Maroon" title="c6">o</font>  st <font color="Olive" title="b1">B</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">C</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">A</font> holds <br/><font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">C</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:24_1_1_2"/><span class="lab"><font color="Green" title="E23">A11</font></span>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">o2</font>,<font color="Maroon" title="c6">o</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">B</font>, <font color="Olive" title="b2">C</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">o2</font>,<font color="Maroon" title="c6">o</font>  st <font color="Olive" title="b1">B</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">C</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">A</font> holds <br/><font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">C</font></span><br/>

<a NAME="E3:24_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A12</font></span>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">o1</font>,<font color="Maroon" title="c6">o</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="altcat_1.html#D4" title="ALTCAT_1:def.4">ALTCAT_1:def 4</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c7">B</font>, <font color="Maroon" title="c8">C</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">o2</font>,<font color="Maroon" title="c6">o</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">B</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">C</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">A</font> implies <font color="Maroon" title="c7">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">C</font> )</span><br/>

<a NAME="E4:24_1_1_2"/><span class="lab"><font color="Green" title="E25">A13</font></span>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">o2</font>,<font color="Maroon" title="c6">o</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c3">o2</font>,<font color="Maroon" title="c6">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="altcat_1.html#D16" title="ALTCAT_1:def.16">ALTCAT_1:def 16</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">B1</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E6:24_1_1_2"/><span class="lab"><font color="Green" title="E26">A14</font></span>: 
<font color="Maroon" title="c9">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">B</font>
 <span class="kw">and </span><br/><a NAME="E7:24_1_1_2"/><span class="lab"><font color="Green" title="E27">A15</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">o2</font>
 <span class="kw">and </span><br/><a NAME="E8:24_1_1_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">B1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_1_1_2"><span class="lab"><font color="Green" title="E23">A11</font></span></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">C1</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E10:24_1_1_2"/><span class="lab"><font color="Green" title="E28">A16</font></span>: 
<font color="Maroon" title="c10">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">C</font>
 <span class="kw">and </span><br/><a NAME="E11:24_1_1_2"/><span class="lab"><font color="Green" title="E29">A17</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c10">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">o2</font>
 <span class="kw">and </span><br/><a NAME="E12:24_1_1_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">C1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_1_1_2"><span class="lab"><font color="Green" title="E23">A11</font></span></a>, <a class="txt" href="#E4:24_1_1_2"><span class="lab"><font color="Green" title="E25">A13</font></span></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></span>;<br/>
<span class="kw">assume </span><a NAME="E13:24_1_1_2"/>
<font color="Maroon" title="c7">B</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">C</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">C</font></span><br/>

<span class="kw">then </span><span class="lab"><font color="Green" title="E30">A18</font></span>: <font color="Maroon" title="c9">B1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c5">F</font> = 
<font color="Maroon" title="c8">C</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">A</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E2:24"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="txt" href="#E2:24_1_1_2"><span class="lab"><font color="Green" title="E23">A11</font></span></a>, <a class="txt" href="#E6:24_1_1_2"><span class="lab"><font color="Green" title="E26">A14</font></span></a>, <a class="txt" href="#E3:24_1_1_2"><span class="lab"><font color="Green" title="E24">A12</font></span></a>, <a class="ref" href="altcat_1.html#T18" title="ALTCAT_1:th.18">ALTCAT_1:18</a></span>
<br/>.= 
<font color="Maroon" title="c10">C1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c5">F</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E2:24"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="txt" href="#E2:24_1_1_2"><span class="lab"><font color="Green" title="E23">A11</font></span></a>, <a class="txt" href="#E10:24_1_1_2"><span class="lab"><font color="Green" title="E28">A16</font></span></a>, <a class="txt" href="#E3:24_1_1_2"><span class="lab"><font color="Green" title="E24">A12</font></span></a>, <a class="ref" href="altcat_1.html#T18" title="ALTCAT_1:th.18">ALTCAT_1:18</a></span>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:24_1_1_2_2"/>
<font color="Maroon" title="c9">B1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">C1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:24_1_1_2_2"/><span class="lab"><font color="Green" title="E31">A19</font></span>: 
<font color="Maroon" title="c11">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">o2</font>
 <span class="kw">and </span><br/><a NAME="E4:24_1_1_2_2"/><span class="lab"><font color="Green" title="E32">A20</font></span>: 
<font color="Maroon" title="c9">B1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">C1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:24_1_1_2"><span class="lab"><font color="Green" title="E27">A15</font></span></a>, <a class="txt" href="#E11:24_1_1_2"><span class="lab"><font color="Green" title="E29">A17</font></span></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></span>;<br/><a NAME="E5:24_1_1_2_2"/>
<font color="Maroon" title="c11">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1_1_2"><span class="lab"><font color="Green" title="E22">A10</font></span></a>, <a class="txt" href="#E3:24_1_1_2_2"><span class="lab"><font color="Green" title="E31">A19</font></span></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E7:24_1_1_2_2"/><span class="lab"><font color="Green" title="E33">A21</font></span>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">F</font>
 <span class="kw">and </span><br/><a NAME="E8:24_1_1_2_2"/><span class="lab"><font color="Green" title="E34">A22</font></span>: 
<font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">z</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><a NAME="E9:24_1_1_2_2"/>
<font color="Maroon" title="c9">B1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">B1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:24_1_1_2_2"><span class="lab"><font color="Green" title="E33">A21</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>;<br/><span class="kw">hence </span><a NAME="E10:24_1_1_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:24_1_1_2"><span class="lab"><font color="Green" title="E30">A18</font></span></a>, <a class="txt" href="#E4:24_1_1_2_2"><span class="lab"><font color="Green" title="E32">A20</font></span></a>, <a class="txt" href="#E7:24_1_1_2_2"><span class="lab"><font color="Green" title="E33">A21</font></span></a>, <a class="txt" href="#E8:24_1_1_2_2"><span class="lab"><font color="Green" title="E34">A22</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E16:24_1_1_2"/>
<font color="Maroon" title="c7">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:24_1_1_2"><span class="lab"><font color="Green" title="E26">A14</font></span></a>, <a class="txt" href="#E10:24_1_1_2"><span class="lab"><font color="Green" title="E28">A16</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_2"><span class="kw">suppose </span></a><a NAME="E1:24_1_2"/><span class="lab"><font color="Green" title="E21">A23</font></span>: 
 ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> &amp; not <font color="Olive" title="b1">x</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">A</font> is  <a href="altcat_3.html#V5" title="ALTCAT_3:attr.5">epi</a>  iff <font color="Maroon" title="c5">F</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  )</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:24_1_2_1_1"/>
( <font color="Maroon" title="c3">o2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c3">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:24_1_2_1_1_1"/><span class="lab"><font color="Green" title="E22">A24</font></span>: 
<font color="Maroon" title="c3">o2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c4">A</font> is  <a href="altcat_3.html#V5" title="ALTCAT_3:attr.5">epi</a>  implies <font color="Maroon" title="c5">F</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  ) &amp; ( <font color="Maroon" title="c5">F</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  implies <font color="Maroon" title="c4">A</font> is  <a href="altcat_3.html#V5" title="ALTCAT_3:attr.5">epi</a>  ) )</span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c6">o</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:24_1_2_1_1_1"/><span class="lab"><font color="Green" title="E23">A25</font></span>: 
<font color="Maroon" title="c6">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 <span class="kw">and </span><br/><a NAME="E4:24_1_2_1_1_1"/><span class="lab"><font color="Green" title="E24">A26</font></span>: 
not <font color="Maroon" title="c6">o</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1_2"><span class="lab"><font color="Green" title="E21">A23</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">o</font> = <font color="Maroon" title="c6">o</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">X</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24_1_2_1_1_1"><span class="lab"><font color="Green" title="E23">A25</font></span></a>, <a class="ref" href="altcat_1.html#D16" title="ALTCAT_1:def.16">ALTCAT_1:def 16</a></span>;<br/><a NAME="E6:24_1_2_1_1_1"/><span class="lab"><font color="Green" title="E25">A27</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">o1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1_2_1_1_1"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/><span class="kw">thus </span><a NAME="E7:24_1_2_1_1_1"/>
( <font color="Maroon" title="c4">A</font> is  <a href="altcat_3.html#V5" title="ALTCAT_3:attr.5">epi</a>  implies <font color="Maroon" title="c5">F</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">F</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  implies <font color="Maroon" title="c4">A</font> is  <a href="altcat_3.html#V5" title="ALTCAT_3:attr.5">epi</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c8">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">o</font>;<br/>
<a NAME="E2:24_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E26">A28</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">o2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">ok</font> = <font color="Maroon" title="c7">o</font> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">k</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as  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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24_1_2_1_1_1"><span class="lab"><font color="Green" title="E24">A26</font></span></a>, <a class="ref" href="realset1.html#T4" title="REALSET1:th.4">REALSET1:4</a></span>;<br/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E4:24_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E27">A29</font></span>: 
<font color="Maroon" title="c4">A</font> is  <a href="altcat_3.html#V5" title="ALTCAT_3:attr.5">epi</a> 
 <span class="kw">and </span><br/><a NAME="E5:24_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E28">A30</font></span>: 
not <font color="Maroon" title="c5">F</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E6:24_1_2_1_1_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">F</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">o2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:24_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E28">A30</font></span></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/>
<a NAME="E7:24_1_2_1_1_1_1"/><span class="kw">then </span>
not <font color="Maroon" title="c3">o2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E26">A28</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E9:24_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E29">A31</font></span>: 
<font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">o2</font>
 <span class="kw">and </span><br/><a NAME="E10:24_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E30">A32</font></span>: 
not <font color="Maroon" title="c10">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">F</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">C</font> = <font color="Maroon" title="c3">o2</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c8">k</font>;<br/>
<a NAME="E11:24_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E31">A33</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">o2</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">o2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></span>;<br/>
<a NAME="E12:24_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E32">A34</font></span>: 
<font color="Maroon" title="c7">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:24_1_2_1_1_1"><span class="lab"><font color="Green" title="E23">A25</font></span></a></span>;<br/>
<a NAME="E13:24_1_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A35</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">o</font>
 
;<br/>
<a NAME="E14:24_1_2_1_1_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">o2</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">o</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2_1_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">o2</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c8">k</font></span>)</span> or <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">o</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:24_1_2_1_1_1_1_1"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">o2</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c8">k</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">o</font></span><br/>

<a NAME="E2:24_1_2_1_1_1_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">o2</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c8">k</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">o2</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c8">k</font></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> <font color="Maroon" title="c12">y</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:24_1_2_1_1_1_1_1"/>
<font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:24_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E33">A35</font></span></a>, <a class="txt" href="#E11:24_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E31">A33</font></span></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E15:24_1_2_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">o2</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c8">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c3">o2</font>,<font color="Maroon" title="c7">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:24_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E31">A33</font></span></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">C1</font> = <font color="Maroon" title="c3">o2</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c8">k</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">o2</font>,<font color="Maroon" title="c7">o</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="altcat_1.html#D16" title="ALTCAT_1:def.16">ALTCAT_1:def 16</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">l</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c9">ok</font>;<br/>
<a NAME="E18:24_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E34">A36</font></span>: 
not <font color="Maroon" title="c13">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">k</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">l</font> = <font color="Maroon" title="c13">l</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">o</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E20:24_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E35">A37</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">l</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:24_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E34">A36</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c7">o</font> =  <a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> $1,<font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c14">l</font>,<font color="Maroon" title="c8">k</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c15">B</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E22:24_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E36">A38</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c15">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">o2</font>
 <span class="kw">and </span><br/><a NAME="E23:24_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E37">A39</font></span>: 
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="c3">o2</font> holds <br/><font color="Maroon" title="c15">B</font> <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> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_1.html#S3" title="FUNCT_1:sch.3">FUNCT_1:sch 3</a>();<br/></span>
<a NAME="E24:24_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E38">A40</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">B</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">o1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:24_1_2_1_1_1"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="txt" href="#E2:24_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E26">A28</font></span></a>, <a class="txt" href="#E22:24_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E36">A38</font></span></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></span>;<br/>
<a NAME="E25:24_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E39">A41</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">o</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2_1_1_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">y1</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c16">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">B</font> or <font color="Maroon" title="c16">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">o</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:24_1_2_1_1_1_1_2"/>
<font color="Maroon" title="c16">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">B</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">o</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:24_1_2_1_1_1_1_2"/><span class="lab"><font color="Green" title="E40">A42</font></span>: 
( <font color="Maroon" title="c17">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c15">B</font> &amp; <font color="Maroon" title="c15">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">y1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_2_1_1_1_1_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:24_1_2_1_1_1_1_2_1"/>
( <font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y</font> or <font color="Maroon" title="c17">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">y</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_2_1_1_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:24_1_2_1_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E41">A43</font></span>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">o</font></span><br/><div class="add"><font color="Maroon" title="c16">y1</font> = 
 <a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> <font color="Maroon" title="c17">x</font>,<font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c14">l</font>,<font color="Maroon" title="c8">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:24_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E36">A38</font></span></a>, <a class="txt" href="#E23:24_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E37">A39</font></span></a>, <a class="txt" href="#E3:24_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E40">A42</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c14">l</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1_2_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E41">A43</font></span></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:24_1_2_1_1_1_1_2_1_1"/>
<font color="Maroon" title="c16">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:24_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E32">A34</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_2_1_1_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:24_1_2_1_1_1_1_2_1_2"/><span class="lab"><font color="Green" title="E41">A44</font></span>: 
<font color="Maroon" title="c17">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">o</font></span><br/><div class="add"><font color="Maroon" title="c16">y1</font> = 
 <a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> <font color="Maroon" title="c17">x</font>,<font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c14">l</font>,<font color="Maroon" title="c8">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:24_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E36">A38</font></span></a>, <a class="txt" href="#E23:24_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E37">A39</font></span></a>, <a class="txt" href="#E3:24_1_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E40">A42</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c8">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1_2_1_1_1_1_2_1_2"><span class="lab"><font color="Green" title="E41">A44</font></span></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:24_1_2_1_1_1_1_2_1_2"/>
<font color="Maroon" title="c16">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:24_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E32">A34</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E26:24_1_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A45</font></span>: 
<font color="Maroon" title="c15">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c3">o2</font>,<font color="Maroon" title="c7">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:24_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E36">A38</font></span></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></span>;<br/>
<a NAME="E27:24_1_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A46</font></span>: 
<font color="Maroon" title="c15">B</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">o2</font>,<font color="Maroon" title="c7">o</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="altcat_1.html#D16" title="ALTCAT_1:def.16">ALTCAT_1:def 16</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">B1</font> = <font color="Maroon" title="c15">B</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">o2</font>,<font color="Maroon" title="c7">o</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:24_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E40">A45</font></span></a>, <a class="ref" href="altcat_1.html#D16" title="ALTCAT_1:def.16">ALTCAT_1:def 16</a></span>;<br/>
<a NAME="E29:24_1_2_1_1_1_1"/>
for <font color="Olive" title="b1">z</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">B</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c5">F</font></span>)</span> holds <br/><font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T25" title="FUNCT_1:th.25">FUNCT_1:25</a></span>;<br/>
<a NAME="E30:24_1_2_1_1_1_1"/><span class="kw">then </span>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">B</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c15">B</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<a NAME="E31:24_1_2_1_1_1_1"/><span class="kw">then </span>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">B</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:24_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E39">A41</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E32:24_1_2_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">B</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c5">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c2">o1</font>,<font color="Maroon" title="c7">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:24_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E38">A40</font></span></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></span>;<br/>
<a NAME="E33:24_1_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A47</font></span>: 
<font color="Maroon" title="c15">B</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c5">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">o1</font>,<font color="Maroon" title="c7">o</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="altcat_1.html#D16" title="ALTCAT_1:def.16">ALTCAT_1:def 16</a></span>;<br/>
<font color="Maroon" title="c15">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">y</font> = 
 <a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> <font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c14">l</font>,<font color="Maroon" title="c8">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:24_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E29">A31</font></span></a>, <a class="txt" href="#E23:24_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E37">A39</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c14">l</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></span>
;<br/>
<a NAME="E35:24_1_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A48</font></span>: 
not <font color="Maroon" title="c15">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">o2</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c8">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:24_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E29">A31</font></span></a>, <a class="txt" href="#E20:24_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E35">A37</font></span></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/>
<a NAME="E36:24_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E44">A49</font></span>: 
 <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"><font color="Maroon" title="c3">o2</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">o1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:24_1_2_1_1_1"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="txt" href="#E2:24_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E26">A28</font></span></a>, <a class="txt" href="#E11:24_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E31">A33</font></span></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2_1_1_1_1_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c17">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">o1</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c15">B</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">o2</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font> )</span><br/><span class="kw">assume </span><a NAME="E1:24_1_2_1_1_1_1_4"/><span class="lab"><font color="Green" title="E45">A50</font></span>: 
<font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">o1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c15">B</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">o2</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font></span><br/><a NAME="E2:24_1_2_1_1_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A51</font></span>: 
<font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:24_1_2_1_1_1"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A52</font></span>: <font color="Maroon" title="c15">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font></span>)</span> = 
 <a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font></span>)</span>,<font color="Maroon" title="c10">y</font>,<font color="Maroon" title="c14">l</font>,<font color="Maroon" title="c8">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E26">A28</font></span></a>, <a class="txt" href="#E23:24_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E37">A39</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c8">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:24_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E30">A32</font></span></a>, <a class="txt" href="#E2:24_1_2_1_1_1_1_4"><span class="lab"><font color="Green" title="E46">A51</font></span></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></span>
;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c15">B</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font> = 
<font color="Maroon" title="c15">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:24_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E38">A40</font></span></a>, <a class="txt" href="#E1:24_1_2_1_1_1_1_4"><span class="lab"><font color="Green" title="E45">A50</font></span></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">o2</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E26">A28</font></span></a>, <a class="txt" href="#E2:24_1_2_1_1_1_1_4"><span class="lab"><font color="Green" title="E46">A51</font></span></a>, <a class="txt" href="#E3:24_1_2_1_1_1_1_4"><span class="lab"><font color="Green" title="E47">A52</font></span></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">o2</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:24_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E44">A49</font></span></a>, <a class="txt" href="#E1:24_1_2_1_1_1_1_4"><span class="lab"><font color="Green" title="E45">A50</font></span></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E38:24_1_2_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">B</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">o2</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c5">F</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:24_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E38">A40</font></span></a>, <a class="txt" href="#E36:24_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E44">A49</font></span></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></span>;<br/>
<span class="kw">then </span><font color="Maroon" title="c16">B1</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">A</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">o2</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c5">F</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E2:24"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="txt" href="#E27:24_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E41">A46</font></span></a>, <a class="txt" href="#E33:24_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E42">A47</font></span></a>, <a class="ref" href="altcat_1.html#T18" title="ALTCAT_1:th.18">ALTCAT_1:18</a></span>
<br/>.= 
<font color="Maroon" title="c12">C1</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">A</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E2:24"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="txt" href="#E27:24_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E41">A46</font></span></a>, <a class="txt" href="#E33:24_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E42">A47</font></span></a>, <a class="ref" href="altcat_1.html#T18" title="ALTCAT_1:th.18">ALTCAT_1:18</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E40:24_1_2_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:24_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="txt" href="#E35:24_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E43">A48</font></span></a>, <a class="txt" href="#E27:24_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E41">A46</font></span></a>, <a class="ref" href="altcat_3.html#D8" target="_self" title="ALTCAT_3:def.8">Def8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">thus </span><a NAME="E8:24_1_2_1_1_1"/>
( <font color="Maroon" title="c5">F</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  implies <font color="Maroon" title="c4">A</font> is  <a href="altcat_3.html#V5" title="ALTCAT_3:attr.5">epi</a>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2_1_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:24_1_2_1_1_1_2"/><span class="lab"><font color="Green" title="E26">A53</font></span>: 
<font color="Maroon" title="c5">F</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">A</font> is  <a href="altcat_3.html#V5" title="ALTCAT_3:attr.5">epi</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c8">o</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">X</font></span>)</span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="altcat_3.html#D8" target="_self" title="ALTCAT_3:def.8">ALTCAT_3:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">o2</font>,<font color="Maroon" title="c8">o</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies for <font color="Olive" title="b1">B</font>, <font color="Olive" title="b2">C</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">o2</font>,<font color="Maroon" title="c8">o</font>  st <font color="Olive" title="b1">B</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">C</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">A</font> holds <br/><font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">C</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:24_1_2_1_1_1_2"/><span class="lab"><font color="Green" title="E27">A54</font></span>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">o2</font>,<font color="Maroon" title="c8">o</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">B</font>, <font color="Olive" title="b2">C</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">o2</font>,<font color="Maroon" title="c8">o</font>  st <font color="Olive" title="b1">B</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">C</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">A</font> holds <br/><font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">C</font></span><br/>

<a NAME="E3:24_1_2_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A55</font></span>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">o1</font>,<font color="Maroon" title="c8">o</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="altcat_1.html#D4" title="ALTCAT_1:def.4">ALTCAT_1:def 4</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c9">B</font>, <font color="Maroon" title="c10">C</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">o2</font>,<font color="Maroon" title="c8">o</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">B</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">C</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">A</font> implies <font color="Maroon" title="c9">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">C</font> )</span><br/>

<a NAME="E4:24_1_2_1_1_1_2"/><span class="lab"><font color="Green" title="E29">A56</font></span>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">o2</font>,<font color="Maroon" title="c8">o</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c3">o2</font>,<font color="Maroon" title="c8">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="altcat_1.html#D16" title="ALTCAT_1:def.16">ALTCAT_1:def 16</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">B1</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E6:24_1_2_1_1_1_2"/><span class="lab"><font color="Green" title="E30">A57</font></span>: 
<font color="Maroon" title="c11">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">B</font>
 <span class="kw">and </span><br/><a NAME="E7:24_1_2_1_1_1_2"/><span class="lab"><font color="Green" title="E31">A58</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c11">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">o2</font>
 <span class="kw">and </span><br/><a NAME="E8:24_1_2_1_1_1_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">B1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E27">A54</font></span></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">C1</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E10:24_1_2_1_1_1_2"/><span class="lab"><font color="Green" title="E32">A59</font></span>: 
<font color="Maroon" title="c12">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">C</font>
 <span class="kw">and </span><br/><a NAME="E11:24_1_2_1_1_1_2"/><span class="lab"><font color="Green" title="E33">A60</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c12">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">o2</font>
 <span class="kw">and </span><br/><a NAME="E12:24_1_2_1_1_1_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">C1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E27">A54</font></span></a>, <a class="txt" href="#E4:24_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E29">A56</font></span></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></span>;<br/>
<span class="kw">assume </span><a NAME="E13:24_1_2_1_1_1_2"/>
<font color="Maroon" title="c9">B</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">C</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">C</font></span><br/>

<span class="kw">then </span><span class="lab"><font color="Green" title="E34">A61</font></span>: <font color="Maroon" title="c11">B1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c5">F</font> = 
<font color="Maroon" title="c10">C</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">A</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E2:24"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="txt" href="#E2:24_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E27">A54</font></span></a>, <a class="txt" href="#E6:24_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E30">A57</font></span></a>, <a class="txt" href="#E3:24_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E28">A55</font></span></a>, <a class="ref" href="altcat_1.html#T18" title="ALTCAT_1:th.18">ALTCAT_1:18</a></span>
<br/>.= 
<font color="Maroon" title="c12">C1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c5">F</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E2:24"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="txt" href="#E2:24_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E27">A54</font></span></a>, <a class="txt" href="#E10:24_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E32">A59</font></span></a>, <a class="txt" href="#E3:24_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E28">A55</font></span></a>, <a class="ref" href="altcat_1.html#T18" title="ALTCAT_1:th.18">ALTCAT_1:18</a></span>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2_1_1_1_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:24_1_2_1_1_1_2_2"/>
<font color="Maroon" title="c11">B1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">C1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:24_1_2_1_1_1_2_2"/><span class="lab"><font color="Green" title="E35">A62</font></span>: 
<font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">o2</font>
 <span class="kw">and </span><br/><a NAME="E4:24_1_2_1_1_1_2_2"/><span class="lab"><font color="Green" title="E36">A63</font></span>: 
<font color="Maroon" title="c11">B1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">C1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:24_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E31">A58</font></span></a>, <a class="txt" href="#E11:24_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E33">A60</font></span></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></span>;<br/><a NAME="E5:24_1_2_1_1_1_2_2"/>
<font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E26">A53</font></span></a>, <a class="txt" href="#E3:24_1_2_1_1_1_2_2"><span class="lab"><font color="Green" title="E35">A62</font></span></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E7:24_1_2_1_1_1_2_2"/><span class="lab"><font color="Green" title="E37">A64</font></span>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">F</font>
 <span class="kw">and </span><br/><a NAME="E8:24_1_2_1_1_1_2_2"/><span class="lab"><font color="Green" title="E38">A65</font></span>: 
<font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">z</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/><a NAME="E9:24_1_2_1_1_1_2_2"/>
<font color="Maroon" title="c11">B1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">B1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:24_1_2_1_1_1_2_2"><span class="lab"><font color="Green" title="E37">A64</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>;<br/><span class="kw">hence </span><a NAME="E10:24_1_2_1_1_1_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:24_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E34">A61</font></span></a>, <a class="txt" href="#E4:24_1_2_1_1_1_2_2"><span class="lab"><font color="Green" title="E36">A63</font></span></a>, <a class="txt" href="#E7:24_1_2_1_1_1_2_2"><span class="lab"><font color="Green" title="E37">A64</font></span></a>, <a class="txt" href="#E8:24_1_2_1_1_1_2_2"><span class="lab"><font color="Green" title="E38">A65</font></span></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E16:24_1_2_1_1_1_2"/>
<font color="Maroon" title="c9">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:24_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E30">A57</font></span></a>, <a class="txt" href="#E10:24_1_2_1_1_1_2"><span class="lab"><font color="Green" title="E32">A59</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:24_1_2_1_1_2"/><span class="lab"><font color="Green" title="E22">A66</font></span>: 
<font color="Maroon" title="c3">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c4">A</font> is  <a href="altcat_3.html#V5" title="ALTCAT_3:attr.5">epi</a>  implies <font color="Maroon" title="c5">F</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  ) &amp; ( <font color="Maroon" title="c5">F</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  implies <font color="Maroon" title="c4">A</font> is  <a href="altcat_3.html#V5" title="ALTCAT_3:attr.5">epi</a>  ) )</span><br/><div class="add"><a NAME="E2:24_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><span class="kw">hence </span><a NAME="E3:24_1_2_1_1_2"/>
( <font color="Maroon" title="c4">A</font> is  <a href="altcat_3.html#V5" title="ALTCAT_3:attr.5">epi</a>  implies <font color="Maroon" title="c5">F</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1_2_1_1_2"><span class="lab"><font color="Green" title="E22">A66</font></span></a>, <a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a>, <a class="ref" href="relat_1.html#T60" title="RELAT_1:th.60">RELAT_1:60</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">F</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  implies <font color="Maroon" title="c4">A</font> is  <a href="altcat_3.html#V5" title="ALTCAT_3:attr.5">epi</a>  )</span><br/><span class="kw">thus </span><a NAME="E4:24_1_2_1_1_2"/>
( <font color="Maroon" title="c5">F</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  implies <font color="Maroon" title="c4">A</font> is  <a href="altcat_3.html#V5" title="ALTCAT_3:attr.5">epi</a>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:24_1_2_1_1_2_1"/>
<font color="Maroon" title="c5">F</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">A</font> is  <a href="altcat_3.html#V5" title="ALTCAT_3:attr.5">epi</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c6">o</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <span class="p1">(<span class="default"><a href="altcat_1.html#K7" title="ALTCAT_1:func.7">EnsCat</a> <font color="Maroon" title="c1">X</font></span>)</span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="altcat_3.html#D8" target="_self" title="ALTCAT_3:def.8">ALTCAT_3:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">o2</font>,<font color="Maroon" title="c6">o</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies for <font color="Olive" title="b1">B</font>, <font color="Olive" title="b2">C</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">o2</font>,<font color="Maroon" title="c6">o</font>  st <font color="Olive" title="b1">B</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">C</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">A</font> holds <br/><font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">C</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:24_1_2_1_1_2_1"/><span class="lab"><font color="Green" title="E23">A67</font></span>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">o2</font>,<font color="Maroon" title="c6">o</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">B</font>, <font color="Olive" title="b2">C</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">o2</font>,<font color="Maroon" title="c6">o</font>  st <font color="Olive" title="b1">B</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">C</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">A</font> holds <br/><font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">C</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c7">B</font>, <font color="Maroon" title="c8">C</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">o2</font>,<font color="Maroon" title="c6">o</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">B</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">C</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">A</font> implies <font color="Maroon" title="c7">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">C</font> )</span><br/>

<a NAME="E3:24_1_2_1_1_2_1"/><span class="lab"><font color="Green" title="E24">A68</font></span>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">o2</font>,<font color="Maroon" title="c6">o</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon" title="c3">o2</font>,<font color="Maroon" title="c6">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="altcat_1.html#D16" title="ALTCAT_1:def.16">ALTCAT_1:def 16</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">B1</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E5:24_1_2_1_1_2_1"/><span class="lab"><font color="Green" title="E25">A69</font></span>: 
<font color="Maroon" title="c9">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">B</font>
 <span class="kw">and </span><br/><a NAME="E6:24_1_2_1_1_2_1"/><span class="lab"><font color="Green" title="E26">A70</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">o2</font>
 <span class="kw">and </span><br/><a NAME="E7:24_1_2_1_1_2_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">B1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E23">A67</font></span></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></span>;<br/>
<a NAME="E8:24_1_2_1_1_2_1"/><span class="lab"><font color="Green" title="E27">A71</font></span>: 
 ex <font color="Olive" title="b1">C1</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Olive" title="b1">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">C</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">C1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">o2</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">C1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">o</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:24_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E23">A67</font></span></a>, <a class="txt" href="#E3:24_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E24">A68</font></span></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></span>;<br/>
<span class="kw">assume </span><a NAME="E9:24_1_2_1_1_2_1"/>
<font color="Maroon" title="c7">B</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">C</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">A</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">C</font></span><br/>

<a NAME="E10:24_1_2_1_1_2_1"/>
<font color="Maroon" title="c9">B1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1_2_1_1_2"><span class="lab"><font color="Green" title="E22">A66</font></span></a>, <a class="txt" href="#E6:24_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E26">A70</font></span></a>, <a class="ref" href="relat_1.html#T64" title="RELAT_1:th.64">RELAT_1:64</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:24_1_2_1_1_2_1"/>
<font color="Maroon" title="c7">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:24_1_2_1_1_2"><span class="lab"><font color="Green" title="E22">A66</font></span></a>, <a class="txt" href="#E5:24_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E25">A69</font></span></a>, <a class="txt" href="#E8:24_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E27">A71</font></span></a>, <a class="ref" href="relat_1.html#T64" title="RELAT_1:th.64">RELAT_1:64</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:24_1_2"/>
( <font color="Maroon" title="c4">A</font> is  <a href="altcat_3.html#V5" title="ALTCAT_3:attr.5">epi</a>  iff <font color="Maroon" title="c5">F</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
