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<span class="kw">let </span><font color="Maroon" title="c1">C</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="cat_8.html#V7" title="CAT_8:attr.7">with_binary_products</a>  <a href="cat_6.html#NM2" title="CAT_6:NM.2">category</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being   <a href="cat_6.html#NM3" title="CAT_6:NM.3">Object</a> of <font color="Maroon" title="c1">C</font> holds  <span class="p1">(<span class="default"><a href="cat_7.html#K3" title="CAT_7:func.3">id-</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="cat_8.html#K7" title="CAT_8:func.7">[x]</a> <span class="p1">(<span class="default"><a href="cat_7.html#K3" title="CAT_7:func.3">id-</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_7.html#K3" title="CAT_7:func.3">id-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Olive" title="b2">b</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="cat_6.html#NM3" title="CAT_6:NM.3">Object</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="cat_7.html#K3" title="CAT_7:func.3">id-</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="cat_8.html#K7" title="CAT_8:func.7">[x]</a> <span class="p1">(<span class="default"><a href="cat_7.html#K3" title="CAT_7:func.3">id-</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_7.html#K3" title="CAT_7:func.3">id-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c3">b</font></span>)</span></span><br/>



<a NAME="E1:116"/><span class="lab"><font color="Green" title="E68">A1</font></span>: 
 <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c3">b</font></span>)</span>,<font color="Maroon" title="c2">a</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_8.html#T42" target="_self" onmouseover="rs('cat_8/T42')" onmouseout="rh()">Th42</a></span>;<br/>
<a NAME="E2:116"/><span class="lab"><font color="Green" title="E69">A2</font></span>: 
(  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c2">a</font>) <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;  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c3">b</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
;<br/>
<span class="lab"><font color="Green" title="E70">A3</font></span>: <span class="p1">(<span class="default"><a href="cat_7.html#K3" title="CAT_7:func.3">id-</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><a href="cat_8.html#K5" title="CAT_8:func.5">pr1</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>)</span>)</span> = 
 <a href="cat_8.html#K5" title="CAT_8:func.5">pr1</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:116"><span class="lab"><font color="Green" title="E68">A1</font></span></a>, <a class="ref" href="cat_7.html#T18" onmouseover="rs('cat_7/T18')" onmouseout="rh()">CAT_7:18</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_8.html#K5" title="CAT_8:func.5">pr1</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>)</span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><a href="cat_7.html#K3" title="CAT_7:func.3">id-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:116"><span class="lab"><font color="Green" title="E68">A1</font></span></a>, <a class="ref" href="cat_7.html#T18" onmouseover="rs('cat_7/T18')" onmouseout="rh()">CAT_7:18</a></span>
;<br/>
<a NAME="E4:116"/><span class="lab"><font color="Green" title="E71">A4</font></span>: 
 <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c3">b</font></span>)</span>,<font color="Maroon" title="c3">b</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_8.html#T42" target="_self" onmouseover="rs('cat_8/T42')" onmouseout="rh()">Th42</a></span>;<br/>
<span class="p1">(<span class="default"><a href="cat_7.html#K3" title="CAT_7:func.3">id-</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><a href="cat_8.html#K6" title="CAT_8:func.6">pr2</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>)</span>)</span> = 
 <a href="cat_8.html#K6" title="CAT_8:func.6">pr2</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:116"><span class="lab"><font color="Green" title="E71">A4</font></span></a>, <a class="ref" href="cat_7.html#T18" onmouseover="rs('cat_7/T18')" onmouseout="rh()">CAT_7:18</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="cat_8.html#K6" title="CAT_8:func.6">pr2</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>)</span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><a href="cat_7.html#K3" title="CAT_7:func.3">id-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c3">b</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:116"><span class="lab"><font color="Green" title="E71">A4</font></span></a>, <a class="ref" href="cat_7.html#T18" onmouseover="rs('cat_7/T18')" onmouseout="rh()">CAT_7:18</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:116"/>
<span class="p1">(<span class="default"><a href="cat_7.html#K3" title="CAT_7:func.3">id-</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="cat_8.html#K7" title="CAT_8:func.7">[x]</a> <span class="p1">(<span class="default"><a href="cat_7.html#K3" title="CAT_7:func.3">id-</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_7.html#K3" title="CAT_7:func.3">id-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c3">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:116"><span class="lab"><font color="Green" title="E69">A2</font></span></a>, <a class="txt" href="#E3:116"><span class="lab"><font color="Green" title="E70">A3</font></span></a>, <a class="ref" href="cat_8.html#D16" target="_self" onmouseover="rs('cat_8/D16')" onmouseout="rh()">Def16</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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