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<div class="add">

<b>let </b><font color="Maroon">C</font> be   <a href="altcat_1.html#NM4">category</a>;<br/><b>let </b><font color="Maroon">D</font> be  non <a href="struct_0.html#V3">empty</a> <a href="altcat_2.html#NM2">subcategory</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">o1</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font>, <font color="Maroon">o2</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">C</font>;<br/><b>let </b><font color="Maroon">p1</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">D</font>, <font color="Maroon">p2</font> be   <a href="altcat_1.html#NM1">object</a> of <font color="Maroon">D</font>;<br/><b>let </b><font color="Maroon">m</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o1</font>,<font color="Maroon">o2</font>;<br/><b>let </b><font color="Maroon">m1</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">o2</font>,<font color="Maroon">o1</font>;<br/><b>let </b><font color="Maroon">n</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">p1</font>,<font color="Maroon">p2</font>;<br/><b>let </b><font color="Maroon">n1</font> be   <a href="altcat_1.html#NM2">Morphism</a> of <font color="Maroon">p2</font>,<font color="Maroon">p1</font>;<br/>















<b>assume </b><a NAME="E1:43"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o1</font> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o2</font> &amp; <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">m1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n1</font> &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <span class="p1"><a href="altcat_1.html#K1">&lt;^</a><span class="default"><font color="Maroon">p2</font>,<font color="Maroon">p1</font></span><a href="altcat_1.html#K1">^&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/>

<b>thus </b><a NAME="E2:43"/>
( <font color="Maroon">m</font> <a href="altcat_3.html#R1">is_left_inverse_of</a> <font color="Maroon">m1</font> iff <font color="Maroon">n</font> <a href="altcat_3.html#R1">is_left_inverse_of</a> <font color="Maroon">n1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:43_1"/>
( <font color="Maroon">m</font> <a href="altcat_3.html#R1">is_left_inverse_of</a> <font color="Maroon">m1</font> implies <font color="Maroon">n</font> <a href="altcat_3.html#R1">is_left_inverse_of</a> <font color="Maroon">n1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:43_1_1"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">m</font> <a href="altcat_3.html#R1">is_left_inverse_of</a> <font color="Maroon">m1</font>
 ;<br/>

<b>thus </b><font color="Maroon">n</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">n1</font> = 
<font color="Maroon">m</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">m1</font>
<b>by </b><i><a class="ref" href="altcat_4.html#T3" target="_self">Th3</a>, <a class="ref" href="altcat_2.html#T33">ALTCAT_2:33</a></i>
<br/>.= 
 <a href="altcat_1.html#K8">idm</a> <font color="Maroon">o2</font>
<b>by </b><i>, <a class="ref" href="altcat_3.html#D1">ALTCAT_3:def 1</a></i>
<br/>.= 
 <a href="altcat_1.html#K8">idm</a> <font color="Maroon">p2</font>
<b>by </b><i><a class="ref" href="altcat_4.html#T3" target="_self">Th3</a>, <a class="ref" href="altcat_2.html#T35">ALTCAT_2:35</a></i>
; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_3.html#D1">ALTCAT_3:def 1</a><br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:43_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">n</font> <a href="altcat_3.html#R1">is_left_inverse_of</a> <font color="Maroon">n1</font>
 ;<br/>

<b>thus </b><font color="Maroon">m</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">m1</font> = 
<font color="Maroon">n</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">n1</font>
<b>by </b><i><a class="ref" href="altcat_4.html#T3" target="_self">Th3</a>, <a class="ref" href="altcat_2.html#T33">ALTCAT_2:33</a></i>
<br/>.= 
 <a href="altcat_1.html#K8">idm</a> <font color="Maroon">p2</font>
<b>by </b><i>, <a class="ref" href="altcat_3.html#D1">ALTCAT_3:def 1</a></i>
<br/>.= 
 <a href="altcat_1.html#K8">idm</a> <font color="Maroon">o2</font>
<b>by </b><i><a class="ref" href="altcat_4.html#T3" target="_self">Th3</a>, <a class="ref" href="altcat_2.html#T35">ALTCAT_2:35</a></i>
; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_3.html#D1">ALTCAT_3:def 1</a><br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:43"/>
( <font color="Maroon">m</font> <a href="altcat_3.html#NR2" target="_self">is_right_inverse_of</a> <font color="Maroon">m1</font> implies <font color="Maroon">n</font> <a href="altcat_3.html#NR2" target="_self">is_right_inverse_of</a> <font color="Maroon">n1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:43_2"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">m</font> <a href="altcat_3.html#NR2" target="_self">is_right_inverse_of</a> <font color="Maroon">m1</font>
 ;<br/>

<b>thus </b><font color="Maroon">n1</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">n</font> = 
<font color="Maroon">m1</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">m</font>
<b>by </b><i><a class="ref" href="altcat_4.html#T3" target="_self">Th3</a>, <a class="ref" href="altcat_2.html#T33">ALTCAT_2:33</a></i>
<br/>.= 
 <a href="altcat_1.html#K8">idm</a> <font color="Maroon">o1</font>
<b>by </b><i>, <a class="ref" href="altcat_3.html#D1">ALTCAT_3:def 1</a></i>
<br/>.= 
 <a href="altcat_1.html#K8">idm</a> <font color="Maroon">p1</font>
<b>by </b><i><a class="ref" href="altcat_4.html#T3" target="_self">Th3</a>, <a class="ref" href="altcat_2.html#T35">ALTCAT_2:35</a></i>
; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_3.html#D1">ALTCAT_3:def 1</a><br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E4:43"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">n</font> <a href="altcat_3.html#NR2" target="_self">is_right_inverse_of</a> <font color="Maroon">n1</font>
 ;<br/>

<b>thus </b><font color="Maroon">m1</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">m</font> = 
<font color="Maroon">n1</font> <a href="altcat_1.html#K5">*</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="altcat_4.html#T3" target="_self">Th3</a>, <a class="ref" href="altcat_2.html#T33">ALTCAT_2:33</a></i>
<br/>.= 
 <a href="altcat_1.html#K8">idm</a> <font color="Maroon">p1</font>
<b>by </b><i>, <a class="ref" href="altcat_3.html#D1">ALTCAT_3:def 1</a></i>
<br/>.= 
 <a href="altcat_1.html#K8">idm</a> <font color="Maroon">o1</font>
<b>by </b><i><a class="ref" href="altcat_4.html#T3" target="_self">Th3</a>, <a class="ref" href="altcat_2.html#T35">ALTCAT_2:35</a></i>
; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_3.html#D1">ALTCAT_3:def 1</a><br/>


</div>
