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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 <font color="Olive" title="b1">o1</font> is <a href="altcat_3.html#V6" title="ALTCAT_3:attr.6">initial</a> &amp; <font color="Olive" title="b2">o2</font> is <a href="altcat_3.html#V6" title="ALTCAT_3:attr.6">initial</a> holds <br/><font color="Olive" title="b1">o1</font>,<font color="Olive" title="b2">o2</font> <a href="altcat_3.html#R2" title="ALTCAT_3:pred.2">are_iso</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"> ( <font color="Maroon" title="c2">o1</font> is <a href="altcat_3.html#V6" title="ALTCAT_3:attr.6">initial</a> &amp; <font color="Maroon" title="c3">o2</font> is <a href="altcat_3.html#V6" title="ALTCAT_3:attr.6">initial</a> implies <font color="Maroon" title="c2">o1</font>,<font color="Maroon" title="c3">o2</font> <a href="altcat_3.html#R2" title="ALTCAT_3:pred.2">are_iso</a>  )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:44"/><i><font color="Green" title="E26">A1</font></i>: 
<font color="Maroon" title="c2">o1</font> is <a href="altcat_3.html#V6" title="ALTCAT_3:attr.6">initial</a>
 <b>and </b><br/><a NAME="E2:44"/><i><font color="Green" title="E27">A2</font></i>: 
<font color="Maroon" title="c3">o2</font> is <a href="altcat_3.html#V6" title="ALTCAT_3:attr.6">initial</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">o1</font>,<font color="Maroon" title="c3">o2</font> <a href="altcat_3.html#R2" title="ALTCAT_3:pred.2">are_iso</a> </span><br/>

<b>consider </b><font color="Maroon" title="c4">M1</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><b> such that </b><br/><a NAME="E4:44"/><i><font color="Green" title="E28">A3</font></i>: 
( <font color="Maroon" title="c4">M1</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="c3">o2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <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> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a> )
 <b>by </b><i><a class="txt" href="altcat_3.html#E1:44"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="altcat_3.html#D9" target="_self" title="ALTCAT_3:def.9">Def9</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">M2</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="c2">o1</font><b> such that </b><br/><a NAME="E6:44"/><i><font color="Green" title="E29">A4</font></i>: 
( <font color="Maroon" title="c5">M2</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="c2">o1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &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> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a> )
 <b>by </b><i><a class="txt" href="altcat_3.html#E2:44"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="altcat_3.html#D9" target="_self" title="ALTCAT_3:def.9">Def9</a></i>;<br/>
<b>thus </b><a NAME="E7:44"/>
( <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>  )
 <b>by </b><i><a class="txt" href="altcat_3.html#E4:44"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="altcat_3.html#E6:44"><i><font color="Green" title="E29">A4</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_3.html#D6" target="_self" title="ALTCAT_3:def.6">ALTCAT_3:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <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> st <font color="Olive" title="b1">A</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a></span><br/>

<a NAME="E8:44"/><i><font color="Green" title="E30">A5</font></i>: 
(  ex <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="c2">o1</font> st <br/>( <font color="Olive" title="b1">M</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="c2">o1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <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="c2">o1</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a> ) &amp;  ex <font color="Olive" title="b1">N</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="c3">o2</font> st <br/>( <font color="Olive" title="b1">N</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="c3">o2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &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="c3">o2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a> ) )
 
<b>by </b><i><a class="txt" href="altcat_3.html#E1:44"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="altcat_3.html#E2:44"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="altcat_3.html#D9" target="_self" title="ALTCAT_3:def.9">Def9</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E10:44"/><i><font color="Green" title="E31">A6</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="c2">o1</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> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="realset1.html#D4" title="REALSET1:def.4">REALSET1:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E12:44"/><i><font color="Green" title="E32">A7</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="c3">o2</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#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>
 <b>by </b><i><a class="txt" href="altcat_3.html#E8:44"><i><font color="Green" title="E30">A5</font></i></a>, <a class="ref" href="realset1.html#D4" title="REALSET1:def.4">REALSET1:def 4</a></i>;<br/>
<a NAME="E13:44"/><i><font color="Green" title="E33">A8</font></i>: 
( <font color="Maroon" title="c4">M1</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c5">M2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font> &amp; <font color="Maroon" title="c5">M2</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">M1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> )
 
<b>by </b><i><a class="txt" href="altcat_3.html#E10:44"><i><font color="Green" title="E31">A6</font></i></a>, <a class="txt" href="altcat_3.html#E12:44"><i><font color="Green" title="E32">A7</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E14:44"/>
(  <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c2">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> &amp;  <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c3">o2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font> )
 
<b>by </b><i><a class="txt" href="altcat_3.html#E10:44"><i><font color="Green" title="E31">A6</font></i></a>, <a class="txt" href="altcat_3.html#E12:44"><i><font color="Green" title="E32">A7</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E15:44"/><b>then </b>
( <font color="Maroon" title="c5">M2</font> <a href="altcat_3.html#R1" title="ALTCAT_3:pred.1">is_left_inverse_of</a> <font color="Maroon" title="c4">M1</font> &amp; <font color="Maroon" title="c5">M2</font> <a href="altcat_3.html#NR2" title="ALTCAT_3:NR.2">is_right_inverse_of</a> <font color="Maroon" title="c4">M1</font> )
 
<b>by </b><i><a class="txt" href="altcat_3.html#E13:44"><i><font color="Green" title="E33">A8</font></i></a>, <a class="ref" href="altcat_3.html#D1" target="_self" title="ALTCAT_3:def.1">Def1</a></i>;<br/>
<a NAME="E16:44"/><b>then </b>
( <font color="Maroon" title="c4">M1</font> is <a href="altcat_3.html#V1" title="ALTCAT_3:attr.1">retraction</a> &amp; <font color="Maroon" title="c4">M1</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#D2" target="_self" title="ALTCAT_3:def.2">Def2</a>, <a class="ref" href="altcat_3.html#D3" target="_self" title="ALTCAT_3:def.3">Def3</a></i>;<br/>
<a NAME="E17:44"/><b>then </b>
<font color="Maroon" title="c4">M1</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>
 
<b>by </b><i><a class="txt" href="altcat_3.html#E4:44"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="altcat_3.html#E6:44"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="altcat_3.html#T6" target="_self" title="ALTCAT_3:th.6">Th6</a></i>;<br/>
<b>hence </b><a NAME="E18:44"/>
 ex <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> st <font color="Olive" title="b1">A</font> is <a href="altcat_3.html#V3" title="ALTCAT_3:attr.3">iso</a>
 ; <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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