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<span class="kw">let </span><font color="Maroon" title="c1">C</font> be   <a href="cat_1.html#NM4" title="CAT_1:NM.4">Category</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">c</font>, <font color="Olive" title="b2">d</font> being  <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font><br/> for <font color="Olive" title="b3">p1</font>, <font color="Olive" title="b4">p2</font>, <font color="Olive" title="b5">f</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font>  st <font color="Olive" title="b1">c</font> <a href="cat_3.html#R2" title="CAT_3:pred.2">is_a_product_wrt</a> <font color="Olive" title="b3">p1</font>,<font color="Olive" title="b4">p2</font> &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">d</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">c</font> &amp; <font color="Olive" title="b5">f</font> is  <a href="cat_1.html#V5" title="CAT_1:attr.5">invertible</a>  holds <br/><font color="Olive" title="b2">d</font> <a href="cat_3.html#R2" title="CAT_3:pred.2">is_a_product_wrt</a> <font color="Olive" title="b3">p1</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b5">f</font>,<font color="Olive" title="b4">p2</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b5">f</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">c</font>, <font color="Maroon" title="c3">d</font> be   <a href="cat_1.html#NM1" title="CAT_1:NM.1">Object</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</font>, <font color="Olive" title="b3">f</font> being  <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font>  st <font color="Maroon" title="c2">c</font> <a href="cat_3.html#R2" title="CAT_3:pred.2">is_a_product_wrt</a> <font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font> &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Olive" title="b3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">d</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Olive" title="b3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c</font> &amp; <font color="Olive" title="b3">f</font> is  <a href="cat_1.html#V5" title="CAT_1:attr.5">invertible</a>  holds <br/><font color="Maroon" title="c3">d</font> <a href="cat_3.html#R2" title="CAT_3:pred.2">is_a_product_wrt</a> <font color="Olive" title="b1">p1</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b3">f</font>,<font color="Olive" title="b2">p2</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Olive" title="b3">f</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">p1</font>, <font color="Maroon" title="c5">p2</font>, <font color="Maroon" title="c6">f</font> be   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">c</font> <a href="cat_3.html#R2" title="CAT_3:pred.2">is_a_product_wrt</a> <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font> &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">d</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c</font> &amp; <font color="Maroon" title="c6">f</font> is  <a href="cat_1.html#V5" title="CAT_1:attr.5">invertible</a>  implies <font color="Maroon" title="c3">d</font> <a href="cat_3.html#R2" title="CAT_3:pred.2">is_a_product_wrt</a> <font color="Maroon" title="c4">p1</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c5">p2</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c6">f</font> )</span><br/>











<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:78"/><span class="lab"><font color="Green" title="E42">A1</font></span>: 
<font color="Maroon" title="c2">c</font> <a href="cat_3.html#R2" title="CAT_3:pred.2">is_a_product_wrt</a> <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font>
 <span class="kw">and </span><br/><a NAME="E2:78"/><span class="lab"><font color="Green" title="E43">A2</font></span>: 
(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">d</font> &amp;  <a href="graph_1.html#K2" title="GRAPH_1:func.2">cod</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c</font> &amp; <font color="Maroon" title="c6">f</font> is  <a href="cat_1.html#V5" title="CAT_1:attr.5">invertible</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">d</font> <a href="cat_3.html#R2" title="CAT_3:pred.2">is_a_product_wrt</a> <font color="Maroon" title="c4">p1</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c5">p2</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c6">f</font></span><br/>

<a NAME="E3:78"/>
(  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c4">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c</font> &amp;  <a href="graph_1.html#K1" title="GRAPH_1:func.1">dom</a> <font color="Maroon" title="c5">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">c</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:78"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="ref" href="cat_3.html#D16" target="_self" title="CAT_3:def.16">Def16</a></span>;<br/>
<a NAME="E4:78"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A3</font></span>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font> is    <a href="cat_3.html#M1" title="CAT_3:mode.1">Projections_family</a> of <font color="Maroon" title="c2">c</font>,<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_3.html#T48" target="_self" title="CAT_3:th.48">Th48</a></span>;<br/>
<a NAME="E5:78"/>
<font color="Maroon" title="c2">c</font> <a href="cat_3.html#R1" title="CAT_3:pred.1">is_a_product_wrt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:78"><span class="lab"><font color="Green" title="E42">A1</font></span></a>, <a class="ref" href="cat_3.html#T59" target="_self" title="CAT_3:th.59">Th59</a></span>;<br/>
<a NAME="E6:78"/><span class="kw">then </span>
<font color="Maroon" title="c3">d</font> <a href="cat_3.html#R1" title="CAT_3:pred.1">is_a_product_wrt</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font></span>)</span> <a href="cat_3.html#K6" title="CAT_3:func.6">*</a> <font color="Maroon" title="c6">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:78"><span class="lab"><font color="Green" title="E43">A2</font></span></a>, <a class="txt" href="#E4:78"><span class="lab"><font color="Green" title="E44">A3</font></span></a>, <a class="ref" href="cat_3.html#T56" target="_self" title="CAT_3:th.56">Th56</a></span>;<br/>
<a NAME="E7:78"/><span class="kw">then </span>
<font color="Maroon" title="c3">d</font> <a href="cat_3.html#R1" title="CAT_3:pred.1">is_a_product_wrt</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1 <a href="funct_4.html#K5" title="FUNCT_4:func.5">--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p1</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c6">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">p2</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c6">f</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_3.html#T18" target="_self" title="CAT_3:th.18">Th18</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:78"/>
<font color="Maroon" title="c3">d</font> <a href="cat_3.html#R2" title="CAT_3:pred.2">is_a_product_wrt</a> <font color="Maroon" title="c4">p1</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c5">p2</font> <a href="cat_1.html#K1" title="CAT_1:func.1">*</a> <font color="Maroon" title="c6">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_3.html#T59" target="_self" title="CAT_3:th.59">Th59</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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