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<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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 <font color="Maroon" title="c1">C</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>  &amp; <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> <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> holds <br/> ( <font color="Olive" title="b3">A</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a> iff ( <font color="Olive" title="b3">A</font> is <a href="altcat_3.html#V1" title="ALTCAT_3:attr.1">retraction</a> &amp; <font color="Olive" title="b3">A</font> is <a href="altcat_3.html#V2" title="ALTCAT_3:attr.2">coretraction</a> ) )</span><br/><b>let </b><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 <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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>  &amp; <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> <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> holds <br/> ( <font color="Olive" title="b1">A</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a> iff ( <font color="Olive" title="b1">A</font> is <a href="altcat_3.html#V1" title="ALTCAT_3:attr.1">retraction</a> &amp; <font color="Olive" title="b1">A</font> is <a href="altcat_3.html#V2" title="ALTCAT_3:attr.2">coretraction</a> ) ) )</span><br/>



<b>assume </b><a NAME="E1:12"/><i><font color="Green" title="E12">A1</font></i>: 
( <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>  &amp; <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> <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:()"><i><font color="Red">::  thesis: </font></i></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> holds <br/> ( <font color="Olive" title="b1">A</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a> iff ( <font color="Olive" title="b1">A</font> is <a href="altcat_3.html#V1" title="ALTCAT_3:attr.1">retraction</a> &amp; <font color="Olive" title="b1">A</font> is <a href="altcat_3.html#V2" title="ALTCAT_3:attr.2">coretraction</a> ) )</span><br/>

<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">A</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a> iff ( <font color="Maroon" title="c4">A</font> is <a href="altcat_3.html#V1" title="ALTCAT_3:attr.1">retraction</a> &amp; <font color="Maroon" title="c4">A</font> is <a href="altcat_3.html#V2" title="ALTCAT_3:attr.2">coretraction</a> ) )</span><br/>

<b>thus </b><a NAME="E2:12"/>
( <font color="Maroon" title="c4">A</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a> implies ( <font color="Maroon" title="c4">A</font> is <a href="altcat_3.html#V1" title="ALTCAT_3:attr.1">retraction</a> &amp; <font color="Maroon" title="c4">A</font> is <a href="altcat_3.html#V2" title="ALTCAT_3:attr.2">coretraction</a> ) )
 <b>by </b><i><a class="ref" href="altcat_3.html#T5" target="_self" title="ALTCAT_3:th.5">Th5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">A</font> is <a href="altcat_3.html#V1" title="ALTCAT_3:attr.1">retraction</a> &amp; <font color="Maroon" title="c4">A</font> is <a href="altcat_3.html#V2" title="ALTCAT_3:attr.2">coretraction</a> implies <font color="Maroon" title="c4">A</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E3:12"/><i><font color="Green" title="E13">A2</font></i>: 
<font color="Maroon" title="c4">A</font> is <a href="altcat_3.html#V1" title="ALTCAT_3:attr.1">retraction</a>
 <b>and </b><br/><a NAME="E4:12"/><i><font color="Green" title="E14">A3</font></i>: 
<font color="Maroon" title="c4">A</font> is <a href="altcat_3.html#V2" title="ALTCAT_3:attr.2">coretraction</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">A</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a></span><br/>

<a NAME="E5:12"/>
( <font color="Maroon" title="c4">A</font> <a href="altcat_3.html#K1" title="ALTCAT_3:func.1">"</a>  <a href="altcat_3.html#R1" title="ALTCAT_3:pred.1">is_left_inverse_of</a> <font color="Maroon" title="c4">A</font> &amp; <font color="Maroon" title="c4">A</font> <a href="altcat_3.html#K1" title="ALTCAT_3:func.1">"</a>  <a href="altcat_3.html#NR2" title="ALTCAT_3:NR.2">is_right_inverse_of</a> <font color="Maroon" title="c4">A</font> )
 
<b>by </b><i><a class="txt" href="#E1:12"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="#E3:12"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="#E4:12"><i><font color="Green" title="E14">A3</font></i></a>, <a class="ref" href="altcat_3.html#D4" target="_self" title="ALTCAT_3:def.4">Def4</a></i>;<br/>
<a NAME="E6:12"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">A</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="c4">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c2">o1</font> &amp; <font color="Maroon" title="c4">A</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">A</font> <a href="altcat_3.html#K1" title="ALTCAT_3:func.1">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c3">o2</font> )
 
<b>by </b><i><a class="ref" href="altcat_3.html#D1" target="_self" title="ALTCAT_3:def.1">Def1</a></i>;<br/>
<b>hence </b><a NAME="E7:12"/>
<font color="Maroon" title="c4">A</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>
 <b>by </b><i><a class="ref" href="altcat_3.html#D5" target="_self" title="ALTCAT_3:def.5">Def5</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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