<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">C</font> be   <a href="altcat_1.html#NM4" title="ALTCAT_1:NM.4">category</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_4.html#K5" title="ALTCAT_4:func.5">AllIso</a> <font color="Maroon" title="c1">C</font></span>)</span><br/> for <font color="Olive" title="b3">m</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>  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/>( <font color="Olive" title="b3">m</font> is  <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>  &amp; <font color="Olive" title="b3">m</font> <a href="altcat_3.html#K1" title="ALTCAT_3:func.1">"</a>  <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="Olive" title="b2">o2</font>,<font color="Olive" title="b1">o1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> )</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_4.html#K5" title="ALTCAT_4:func.5">AllIso</a> <font color="Maroon" title="c1">C</font></span>)</span>; <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">m</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>  st <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>  holds <br/>( <font color="Olive" title="b1">m</font> is  <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>  &amp; <font color="Olive" title="b1">m</font> <a href="altcat_3.html#K1" title="ALTCAT_3:func.1">"</a>  <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="c2">o1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">m</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"> ( <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 ( <font color="Maroon" title="c4">m</font> is  <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>  &amp; <font color="Maroon" title="c4">m</font> <a href="altcat_3.html#K1" title="ALTCAT_3:func.1">"</a>  <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="c2">o1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:67"/><span class="lab"><font color="Green" title="E41">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"> ( <font color="Maroon" title="c4">m</font> is  <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>  &amp; <font color="Maroon" title="c4">m</font> <a href="altcat_3.html#K1" title="ALTCAT_3:func.1">"</a>  <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="c2">o1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">p1</font> = <font color="Maroon" title="c2">o1</font>, <font color="Maroon" title="c6">p2</font> = <font color="Maroon" title="c3">o2</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">C</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="altcat_4.html#D5" target="_self" title="ALTCAT_4:def.5">Def5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">p</font> = <font color="Maroon" title="c4">m</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c5">p1</font>,<font color="Maroon" title="c6">p2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="ref" href="altcat_2.html#T34" title="ALTCAT_2:th.34">ALTCAT_2:34</a></span>;<br/>
<a NAME="E4:67"/>
<font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K5" title="ALTCAT_4:func.5">AllIso</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c2">o1</font>,<font color="Maroon" title="c3">o2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67"><span class="lab"><font color="Green" title="E41">A1</font></span></a></span>;<br/>
<a NAME="E5:67"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A2</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="c5">p1</font>,<font color="Maroon" title="c6">p2</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>  &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c5">p1</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="ref" href="altcat_4.html#D5" target="_self" title="ALTCAT_4:def.5">Def5</a></span>;<br/>
<a NAME="E6:67"/><span class="lab"><font color="Green" title="E43">A3</font></span>: 
<font color="Maroon" title="c7">p</font> is  <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="ref" href="altcat_4.html#D5" target="_self" title="ALTCAT_4:def.5">Def5</a></span>;<br/>
<a NAME="E7:67"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A4</font></span>: 
<font color="Maroon" title="c7">p</font> <a href="altcat_3.html#K1" title="ALTCAT_3:func.1">"</a>  is  <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:67"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="ref" href="altcat_4.html#T3" target="_self" title="ALTCAT_4:th.3">Th3</a></span>;<br/>
<a NAME="E8:67"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A5</font></span>: 
<font color="Maroon" title="c7">p</font> <a href="altcat_3.html#K1" title="ALTCAT_3:func.1">"</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="altcat_1.html#U1" title="ALTCAT_1:sel.1">Arrows</a> of <span class="p1">(<span class="default"><a href="altcat_4.html#K5" title="ALTCAT_4:func.5">AllIso</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">p2</font>,<font color="Maroon" title="c5">p1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:67"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="ref" href="altcat_4.html#D5" target="_self" title="ALTCAT_4:def.5">Def5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">m1</font> = <font color="Maroon" title="c7">p</font> <a href="altcat_3.html#K1" title="ALTCAT_3:func.1">"</a>  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="c2">o1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:67"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="txt" href="#E7:67"><span class="lab"><font color="Green" title="E44">A4</font></span></a>, <a class="ref" href="altcat_4.html#D5" target="_self" title="ALTCAT_4:def.5">Def5</a></span>;<br/>
<a NAME="E10:67"/><span class="lab"><font color="Green" title="E46">A6</font></span>: 
<font color="Maroon" title="c4">m</font> is  <a href="altcat_3.html#V1" title="ALTCAT_3:attr.1">retraction</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span>
<font color="Maroon" title="c8">m1</font>
; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="altcat_3.html#D2" title="ALTCAT_3:def.2">ALTCAT_3:def 2</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">m</font> <a href="altcat_3.html#R1" title="ALTCAT_3:pred.1">is_left_inverse_of</a> <font color="Maroon" title="c8">m1</font></span><br/>

<span class="kw">thus </span><font color="Maroon" title="c4">m</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c8">m1</font> = 
<font color="Maroon" title="c7">p</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="altcat_3.html#K1" title="ALTCAT_3:func.1">"</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E8:67"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="ref" href="altcat_2.html#T33" title="ALTCAT_2:th.33">ALTCAT_2:33</a></span>
<br/>.= 
 <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c6">p2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:67"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="ref" href="altcat_3.html#D5" title="ALTCAT_3:def.5">ALTCAT_3:def 5</a></span>
<br/>.= 
 <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c3">o2</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="altcat_2.html#T35" title="ALTCAT_2:th.35">ALTCAT_2:35</a></span>
; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="altcat_3.html#D1" title="ALTCAT_3:def.1">ALTCAT_3:def 1</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>
<a NAME="E11:67"/><span class="lab"><font color="Green" title="E47">A7</font></span>: 
<font color="Maroon" title="c4">m</font> is  <a href="altcat_3.html#V2" title="ALTCAT_3:attr.2">coretraction</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span>
<font color="Maroon" title="c8">m1</font>
; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="altcat_3.html#D3" title="ALTCAT_3:def.3">ALTCAT_3:def 3</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="c8">m1</font> <a href="altcat_3.html#R1" title="ALTCAT_3:pred.1">is_left_inverse_of</a> <font color="Maroon" title="c4">m</font></span><br/>

<span class="kw">thus </span><font color="Maroon" title="c8">m1</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">m</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="altcat_3.html#K1" title="ALTCAT_3:func.1">"</a> </span>)</span> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c7">p</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E8:67"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="ref" href="altcat_2.html#T33" title="ALTCAT_2:th.33">ALTCAT_2:33</a></span>
<br/>.= 
 <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c5">p1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:67"><span class="lab"><font color="Green" title="E43">A3</font></span></a>, <a class="ref" href="altcat_3.html#D5" title="ALTCAT_3:def.5">ALTCAT_3:def 5</a></span>
<br/>.= 
 <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c2">o1</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="altcat_2.html#T35" title="ALTCAT_2:th.35">ALTCAT_2:35</a></span>
; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="altcat_3.html#D1" title="ALTCAT_3:def.1">ALTCAT_3:def 1</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>
<a NAME="E12:67"/>
<font color="Maroon" title="c7">p</font> <a href="altcat_3.html#K1" title="ALTCAT_3:func.1">"</a>  <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="c2">o1</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="txt" href="#E5:67"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="txt" href="#E7:67"><span class="lab"><font color="Green" title="E44">A4</font></span></a>, <a class="ref" href="altcat_4.html#D5" target="_self" title="ALTCAT_4:def.5">Def5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:67"/>
<font color="Maroon" title="c4">m</font> is  <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:67"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E10:67"><span class="lab"><font color="Green" title="E46">A6</font></span></a>, <a class="txt" href="#E11:67"><span class="lab"><font color="Green" title="E47">A7</font></span></a>, <a class="ref" href="altcat_3.html#T6" title="ALTCAT_3:th.6">ALTCAT_3:6</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="c4">m</font> <a href="altcat_3.html#K1" title="ALTCAT_3:func.1">"</a>  <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="c2">o1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span></span><br/>

<span class="kw">thus </span><a NAME="E14:67"/>
<font color="Maroon" title="c4">m</font> <a href="altcat_3.html#K1" title="ALTCAT_3:func.1">"</a>  <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="c2">o1</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="txt" href="#E8:67"><span class="lab"><font color="Green" title="E45">A5</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>
