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<b>let </b><font color="Maroon" title="c3">a1</font>, <font color="Maroon" title="c4">a2</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font>; <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">o'</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="Maroon" title="c2">o</font>,<font color="Olive" title="b1">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>  holds <br/>for <font color="Olive" title="b2">a</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">o</font>,<font color="Olive" title="b1">o'</font> holds  <font color="Olive" title="b2">a</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c3">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> ) &amp; ( for <font color="Olive" title="b1">o'</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="Maroon" title="c2">o</font>,<font color="Olive" title="b1">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>  holds <br/>for <font color="Olive" title="b2">a</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">o</font>,<font color="Olive" title="b1">o'</font> holds  <font color="Olive" title="b2">a</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> ) implies <font color="Maroon" title="c3">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a2</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:48_1_2"/><i><font color="Green" title="E31">A4</font></i>: 
for <font color="Olive" title="b1">o'</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="Maroon" title="c2">o</font>,<font color="Olive" title="b1">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>  holds <br/>for <font color="Olive" title="b2">a</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">o</font>,<font color="Olive" title="b1">o'</font> holds  <font color="Olive" title="b2">a</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c3">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font>
 <b>and </b><br/><a NAME="E2:48_1_2"/><i><font color="Green" title="E32">A5</font></i>: 
for <font color="Olive" title="b1">o'</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="Maroon" title="c2">o</font>,<font color="Olive" title="b1">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>  holds <br/>for <font color="Olive" title="b2">a</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">o</font>,<font color="Olive" title="b1">o'</font> holds  <font color="Olive" title="b2">a</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a2</font></span><br/>

<a NAME="E3:48_1_2"/>
the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c1">C</font> is <a href="altcat_1.html#V5" title="ALTCAT_1:attr.5">with_left_units</a>
 
<b>by </b><i><a class="ref" href="altcat_1.html#D18" target="_self" title="ALTCAT_1:def.18">Def18</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">d</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:48_1_2"/><i><font color="Green" title="E33">A6</font></i>: 
<font color="Maroon" title="c5">d</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 <font color="Maroon" title="c1">C</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font>
 <b>and </b><br/><a NAME="E6:48_1_2"/><i><font color="Green" title="E34">A7</font></i>: 
for <font color="Olive" title="b1">o'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">C</font><br/> for <font color="Olive" title="b2">f</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">f</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 <font color="Maroon" title="c1">C</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">o'</font>,<font color="Maroon" title="c2">o</font> holds <br/><span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Olive" title="b1">o'</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c5">d</font>,<font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font>
 <b>by </b><i><a class="ref" href="altcat_1.html#D9" target="_self" title="ALTCAT_1:def.9">Def9</a></i>;<br/>
<a NAME="E7:48_1_2"/><i><font color="Green" title="E35">A8</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">o</font>,<font color="Maroon" title="c2">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> 
 
<b>by </b><i><a class="ref" href="altcat_1.html#T21" target="_self" title="ALTCAT_1:th.21">Th21</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">d</font> = <font color="Maroon" title="c5">d</font> as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font> <b>by </b><i><a class="txt" href="#E5:48_1_2"><i><font color="Green" title="E33">A6</font></i></a></i>;<br/>
<b>thus </b><font color="Maroon" title="c3">a1</font> = 
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">d</font>,<font color="Maroon" title="c3">a1</font>
<b>by </b><i><a class="txt" href="#E6:48_1_2"><i><font color="Green" title="E34">A7</font></i></a>, <a class="txt" href="#E7:48_1_2"><i><font color="Green" title="E35">A8</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c6">d</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c3">a1</font>
<b>by </b><i><a class="txt" href="#E7:48_1_2"><i><font color="Green" title="E35">A8</font></i></a>, <a class="ref" href="altcat_1.html#D10" target="_self" title="ALTCAT_1:def.10">Def10</a></i>
<br/>.= 
<font color="Maroon" title="c6">d</font>
<b>by </b><i><a class="txt" href="#E1:48_1_2"><i><font color="Green" title="E31">A4</font></i></a>, <a class="ref" href="altcat_1.html#T21" target="_self" title="ALTCAT_1:th.21">Th21</a></i>
<br/>.= 
<font color="Maroon" title="c6">d</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">a2</font>
<b>by </b><i><a class="txt" href="#E2:48_1_2"><i><font color="Green" title="E32">A5</font></i></a>, <a class="ref" href="altcat_1.html#T21" target="_self" title="ALTCAT_1:th.21">Th21</a></i>
<br/>.= 
<span class="p1">(<span class="default">the <a href="altcat_1.html#U2" title="ALTCAT_1:sel.2">Comp</a> of <font color="Maroon" title="c1">C</font> <a href="altcat_1.html#K4" title="ALTCAT_1:func.4">.</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c6">d</font>,<font color="Maroon" title="c4">a2</font>
<b>by </b><i><a class="txt" href="#E7:48_1_2"><i><font color="Green" title="E35">A8</font></i></a>, <a class="ref" href="altcat_1.html#D10" target="_self" title="ALTCAT_1:def.10">Def10</a></i>
<br/>.= 
<font color="Maroon" title="c4">a2</font>
<b>by </b><i><a class="txt" href="#E6:48_1_2"><i><font color="Green" title="E34">A7</font></i></a>, <a class="txt" href="#E7:48_1_2"><i><font color="Green" title="E35">A8</font></i></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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