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<a NAME="E1:39_1_1"/>
( ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><a href="modcat_1.html#K12" target="_self">LModCat</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> holds <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="cat_1.html#U5">Comp</a> of <span class="p1">(<span class="default"><a href="modcat_1.html#K12" target="_self">LModCat</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> iff  <a href="cat_1.html#K2">dom</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Olive">b<sub>1</sub></font> ) ) &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><a href="modcat_1.html#K12" target="_self">LModCat</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> st  <a href="cat_1.html#K2">dom</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Olive">b<sub>1</sub></font> holds <br/>(  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="cat_1.html#K4">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K2">dom</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="cat_1.html#K4">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Olive">b<sub>2</sub></font> ) ) &amp; ( for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><a href="modcat_1.html#K12" target="_self">LModCat</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> st  <a href="cat_1.html#K2">dom</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Olive">b<sub>2</sub></font> &amp;  <a href="cat_1.html#K2">dom</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="cat_1.html#K3">cod</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>3</sub></font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="cat_1.html#K4">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="cat_1.html#K4">*</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="cat_1.html#NM1">Object</a> of <span class="p1">(<span class="default"><a href="modcat_1.html#K12" target="_self">LModCat</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> holds <br/>(  <a href="cat_1.html#K2">dom</a> <span class="p1">(<span class="default"><a href="cat_1.html#K5">id</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp;  <a href="cat_1.html#K3">cod</a> <span class="p1">(<span class="default"><a href="cat_1.html#K5">id</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><a href="modcat_1.html#K12" target="_self">LModCat</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> st  <a href="cat_1.html#K3">cod</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><span class="p1">(<span class="default"><a href="cat_1.html#K5">id</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="cat_1.html#K4">*</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> ) &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="cat_1.html#NM2">Morphism</a> of <span class="p1">(<span class="default"><a href="modcat_1.html#K12" target="_self">LModCat</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> st  <a href="cat_1.html#K2">dom</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="cat_1.html#K4">*</a> <span class="p1">(<span class="default"><a href="cat_1.html#K5">id</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> ) ) ) )
 
<b>by </b><i><a class="txt" href="modcat_1.html#E34"><i><font color="Green">Lemma27</font></i></a>, <a class="txt" href="modcat_1.html#E35"><i><font color="Green">Lemma28</font></i></a>, <a class="txt" href="modcat_1.html#E36"><i><font color="Green">Lemma29</font></i></a>, <a class="ref" href="modcat_1.html#T14" target="_self">Th14</a></i>;<br/>
<b>hence </b><a NAME="E2:39_1_1"/>
 <a href="modcat_1.html#K12" target="_self">LModCat</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> is <a href="cat_1.html#V2">Category-like</a>
 <b>by </b><i><a class="ref" href="cat_1.html#T29">CAT_1:29</a></i>;<br/>


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