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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>, <font color="Olive" title="b3">c1</font>, <font color="Olive" title="b4">c2</font> being   <a href="cat_6.html#NM3" title="CAT_6:NM.3">Object</a> of <font color="Maroon" title="c1">C</font><br/>  for <font color="Olive" title="b5">e1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Olive" title="b3">c1</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font><br/>  for <font color="Olive" title="b6">e2</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Olive" title="b4">c2</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>  st  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<span class="p1">(<span class="default"><font color="Olive" title="b3">c1</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Olive" title="b1">a</font></span>)</span>,<font color="Olive" title="b2">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>  &amp;  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<span class="p1">(<span class="default"><font color="Olive" title="b4">c2</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Olive" title="b1">a</font></span>)</span>,<font color="Olive" title="b2">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>  &amp; <font color="Olive" title="b3">c1</font>,<font color="Olive" title="b5">e1</font> <a href="cat_8.html#R5" title="CAT_8:pred.5">is_exponent_of</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> &amp; <font color="Olive" title="b4">c2</font>,<font color="Olive" title="b6">e2</font> <a href="cat_8.html#R5" title="CAT_8:pred.5">is_exponent_of</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> holds <br/><font color="Olive" title="b3">c1</font>,<font color="Olive" title="b4">c2</font> <a href="cat_7.html#R1" title="CAT_7:pred.1">are_isomorphic</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">c1</font>, <font color="Maroon" title="c5">c2</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">  for <font color="Olive" title="b1">e1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c4">c1</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font><br/>  for <font color="Olive" title="b2">e2</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c5">c2</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>  st  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">c1</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c2">a</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>  &amp;  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">c2</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c2">a</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>  &amp; <font color="Maroon" title="c4">c1</font>,<font color="Olive" title="b1">e1</font> <a href="cat_8.html#R5" title="CAT_8:pred.5">is_exponent_of</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c5">c2</font>,<font color="Olive" title="b2">e2</font> <a href="cat_8.html#R5" title="CAT_8:pred.5">is_exponent_of</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> holds <br/><font color="Maroon" title="c4">c1</font>,<font color="Maroon" title="c5">c2</font> <a href="cat_7.html#R1" title="CAT_7:pred.1">are_isomorphic</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c6">e1</font> be    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c4">c1</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</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">  for <font color="Olive" title="b1">e2</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c5">c2</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>  st  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">c1</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c2">a</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>  &amp;  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">c2</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c2">a</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>  &amp; <font color="Maroon" title="c4">c1</font>,<font color="Maroon" title="c6">e1</font> <a href="cat_8.html#R5" title="CAT_8:pred.5">is_exponent_of</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c5">c2</font>,<font color="Olive" title="b1">e2</font> <a href="cat_8.html#R5" title="CAT_8:pred.5">is_exponent_of</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> holds <br/><font color="Maroon" title="c4">c1</font>,<font color="Maroon" title="c5">c2</font> <a href="cat_7.html#R1" title="CAT_7:pred.1">are_isomorphic</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c7">e2</font> be    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c5">c2</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</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"> (  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">c1</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c2">a</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>  &amp;  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">c2</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c2">a</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>  &amp; <font color="Maroon" title="c4">c1</font>,<font color="Maroon" title="c6">e1</font> <a href="cat_8.html#R5" title="CAT_8:pred.5">is_exponent_of</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c5">c2</font>,<font color="Maroon" title="c7">e2</font> <a href="cat_8.html#R5" title="CAT_8:pred.5">is_exponent_of</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> implies <font color="Maroon" title="c4">c1</font>,<font color="Maroon" title="c5">c2</font> <a href="cat_7.html#R1" title="CAT_7:pred.1">are_isomorphic</a>  )</span><br/>







<span class="kw">assume </span><a NAME="E1:117"/><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="c4">c1</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c2">a</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> 
 ; <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"> (  not  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c5">c2</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c2">a</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>  or  not <font color="Maroon" title="c4">c1</font>,<font color="Maroon" title="c6">e1</font> <a href="cat_8.html#R5" title="CAT_8:pred.5">is_exponent_of</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> or  not <font color="Maroon" title="c5">c2</font>,<font color="Maroon" title="c7">e2</font> <a href="cat_8.html#R5" title="CAT_8:pred.5">is_exponent_of</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> or <font color="Maroon" title="c4">c1</font>,<font color="Maroon" title="c5">c2</font> <a href="cat_7.html#R1" title="CAT_7:pred.1">are_isomorphic</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E2:117"/><span class="lab"><font color="Green" title="E69">A2</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="c5">c2</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c2">a</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> 
 ; <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"> (  not <font color="Maroon" title="c4">c1</font>,<font color="Maroon" title="c6">e1</font> <a href="cat_8.html#R5" title="CAT_8:pred.5">is_exponent_of</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> or  not <font color="Maroon" title="c5">c2</font>,<font color="Maroon" title="c7">e2</font> <a href="cat_8.html#R5" title="CAT_8:pred.5">is_exponent_of</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> or <font color="Maroon" title="c4">c1</font>,<font color="Maroon" title="c5">c2</font> <a href="cat_7.html#R1" title="CAT_7:pred.1">are_isomorphic</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E3:117"/><span class="lab"><font color="Green" title="E70">A3</font></span>: 
<font color="Maroon" title="c4">c1</font>,<font color="Maroon" title="c6">e1</font> <a href="cat_8.html#R5" title="CAT_8:pred.5">is_exponent_of</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</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"> (  not <font color="Maroon" title="c5">c2</font>,<font color="Maroon" title="c7">e2</font> <a href="cat_8.html#R5" title="CAT_8:pred.5">is_exponent_of</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> or <font color="Maroon" title="c4">c1</font>,<font color="Maroon" title="c5">c2</font> <a href="cat_7.html#R1" title="CAT_7:pred.1">are_isomorphic</a>  )</span><br/>

<a NAME="E4:117"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A4</font></span>: 
(  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<font color="Maroon" title="c5">c2</font>,<font color="Maroon" title="c4">c1</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;  ex <font color="Olive" title="b1">h</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c5">c2</font>,<font color="Maroon" title="c4">c1</font> st <br/>( <font color="Maroon" title="c7">e2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">e1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <a href="cat_8.html#K7" title="CAT_8:func.7">[x]</a> <span class="p2">(<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></span>)</span> &amp; (  for <font color="Olive" title="b2">h1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c5">c2</font>,<font color="Maroon" title="c4">c1</font>  st <font color="Maroon" title="c7">e2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">e1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">h1</font> <a href="cat_8.html#K7" title="CAT_8:func.7">[x]</a> <span class="p2">(<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></span>)</span> holds <br/><font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">h1</font> ) ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:117"><span class="lab"><font color="Green" title="E69">A2</font></span></a>, <a class="txt" href="#E1:117"><span class="lab"><font color="Green" title="E68">A1</font></span></a>, <a class="ref" href="cat_8.html#D29" target="_self" onmouseover="rs('cat_8/D29')" onmouseout="rh()">Def29</a></span>;<br/>
<span class="kw">assume </span><a NAME="E5:117"/><span class="lab"><font color="Green" title="E72">A5</font></span>: 
<font color="Maroon" title="c5">c2</font>,<font color="Maroon" title="c7">e2</font> <a href="cat_8.html#R5" title="CAT_8:pred.5">is_exponent_of</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</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"> <font color="Maroon" title="c4">c1</font>,<font color="Maroon" title="c5">c2</font> <a href="cat_7.html#R1" title="CAT_7:pred.1">are_isomorphic</a> </span><br/>

<a NAME="E6:117"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A6</font></span>: 
(  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<font color="Maroon" title="c4">c1</font>,<font color="Maroon" title="c5">c2</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;  ex <font color="Olive" title="b1">h</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c4">c1</font>,<font color="Maroon" title="c5">c2</font> st <br/>( <font color="Maroon" title="c6">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">e2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <a href="cat_8.html#K7" title="CAT_8:func.7">[x]</a> <span class="p2">(<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></span>)</span> &amp; (  for <font color="Olive" title="b2">h1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c4">c1</font>,<font color="Maroon" title="c5">c2</font>  st <font color="Maroon" title="c6">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">e2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">h1</font> <a href="cat_8.html#K7" title="CAT_8:func.7">[x]</a> <span class="p2">(<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></span>)</span> holds <br/><font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">h1</font> ) ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:117"><span class="lab"><font color="Green" title="E68">A1</font></span></a>, <a class="txt" href="#E2:117"><span class="lab"><font color="Green" title="E69">A2</font></span></a>, <a class="ref" href="cat_8.html#D29" target="_self" onmouseover="rs('cat_8/D29')" onmouseout="rh()">Def29</a></span>;<br/>
<a NAME="E7:117"/>
 ex <font color="Olive" title="b1">f</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c4">c1</font>,<font color="Maroon" title="c5">c2</font> st <font color="Olive" title="b1">f</font> is  <a href="cat_7.html#V5" title="CAT_7:attr.5">isomorphism</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="117_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c8">f</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c4">c1</font>,<font color="Maroon" title="c5">c2</font><span class="kw"> such that </span><br/><a NAME="E2:117_1"/><span class="lab"><font color="Green" title="E74">A7</font></span>: 
( <font color="Maroon" title="c6">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">e2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="cat_8.html#K7" title="CAT_8:func.7">[x]</a> <span class="p2">(<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></span>)</span> &amp; (  for <font color="Olive" title="b1">h1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c4">c1</font>,<font color="Maroon" title="c5">c2</font>  st <font color="Maroon" title="c6">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">e2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">h1</font> <a href="cat_8.html#K7" title="CAT_8:func.7">[x]</a> <span class="p2">(<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></span>)</span> holds <br/><font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h1</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:117"><span class="lab"><font color="Green" title="E68">A1</font></span></a>, <a class="txt" href="#E2:117"><span class="lab"><font color="Green" title="E69">A2</font></span></a>, <a class="txt" href="#E5:117"><span class="lab"><font color="Green" title="E72">A5</font></span></a>, <a class="ref" href="cat_8.html#D29" target="_self" onmouseover="rs('cat_8/D29')" onmouseout="rh()">Def29</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c8">f</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"> <font color="Maroon" title="c8">f</font> is  <a href="cat_7.html#V5" title="CAT_7:attr.5">isomorphism</a> </span><br/>

<a NAME="E3:117_1"/>
 ex <font color="Olive" title="b1">g</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c5">c2</font>,<font color="Maroon" title="c4">c1</font> st <br/>( <font color="Olive" title="b1">g</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_7.html#K3" title="CAT_7:func.3">id-</a> <font color="Maroon" title="c4">c1</font> &amp; <font color="Maroon" title="c8">f</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_7.html#K3" title="CAT_7:func.3">id-</a> <font color="Maroon" title="c5">c2</font> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="117_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c9">g</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c5">c2</font>,<font color="Maroon" title="c4">c1</font><span class="kw"> such that </span><br/><a NAME="E2:117_1_1"/><span class="lab"><font color="Green" title="E75">A8</font></span>: 
( <font color="Maroon" title="c7">e2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">e1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="cat_8.html#K7" title="CAT_8:func.7">[x]</a> <span class="p2">(<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></span>)</span> &amp; (  for <font color="Olive" title="b1">h1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c5">c2</font>,<font color="Maroon" title="c4">c1</font>  st <font color="Maroon" title="c7">e2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">e1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">h1</font> <a href="cat_8.html#K7" title="CAT_8:func.7">[x]</a> <span class="p2">(<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></span>)</span> holds <br/><font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h1</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:117"><span class="lab"><font color="Green" title="E69">A2</font></span></a>, <a class="txt" href="#E1:117"><span class="lab"><font color="Green" title="E68">A1</font></span></a>, <a class="txt" href="#E3:117"><span class="lab"><font color="Green" title="E70">A3</font></span></a>, <a class="ref" href="cat_8.html#D29" target="_self" onmouseover="rs('cat_8/D29')" onmouseout="rh()">Def29</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c9">g</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"> ( <font color="Maroon" title="c9">g</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_7.html#K3" title="CAT_7:func.3">id-</a> <font color="Maroon" title="c4">c1</font> &amp; <font color="Maroon" title="c8">f</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_7.html#K3" title="CAT_7:func.3">id-</a> <font color="Maroon" title="c5">c2</font> )</span><br/>

<a NAME="E3:117_1_1"/><span class="lab"><font color="Green" title="E76">A9</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> 
 
;<br/>
<a NAME="E4:117_1_1"/><span class="lab"><font color="Green" title="E77">A10</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="c4">c1</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c2">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">c2</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c2">a</font></span>)</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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:117_1_1"><span class="lab"><font color="Green" title="E76">A9</font></span></a>, <a class="txt" href="#E6:117"><span class="lab"><font color="Green" title="E73">A6</font></span></a>, <a class="ref" href="cat_8.html#T44" target="_self" onmouseover="rs('cat_8/T44')" onmouseout="rh()">Th44</a></span>;<br/>
<a NAME="E5:117_1_1"/><span class="lab"><font color="Green" title="E78">A11</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="c5">c2</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c2">a</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">c1</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c2">a</font></span>)</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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:117_1_1"><span class="lab"><font color="Green" title="E76">A9</font></span></a>, <a class="txt" href="#E4:117"><span class="lab"><font color="Green" title="E71">A4</font></span></a>, <a class="ref" href="cat_8.html#T44" target="_self" onmouseover="rs('cat_8/T44')" onmouseout="rh()">Th44</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">h2</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c4">c1</font>,<font color="Maroon" title="c4">c1</font><span class="kw"> such that </span><br/><a NAME="E7:117_1_1"/>
<font color="Maroon" title="c6">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">e1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">h2</font> <a href="cat_8.html#K7" title="CAT_8:func.7">[x]</a> <span class="p2">(<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></span>)</span>
 <span class="kw">and </span><br/><a NAME="E8:117_1_1"/><span class="lab"><font color="Green" title="E79">A12</font></span>: 
 for <font color="Olive" title="b1">h1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c4">c1</font>,<font color="Maroon" title="c4">c1</font>  st <font color="Maroon" title="c6">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">e1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">h1</font> <a href="cat_8.html#K7" title="CAT_8:func.7">[x]</a> <span class="p2">(<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></span>)</span> holds <br/><font color="Maroon" title="c10">h2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:117"><span class="lab"><font color="Green" title="E70">A3</font></span></a>, <a class="txt" href="#E1:117"><span class="lab"><font color="Green" title="E68">A1</font></span></a>, <a class="ref" href="cat_8.html#D29" target="_self" onmouseover="rs('cat_8/D29')" onmouseout="rh()">Def29</a></span>;<br/>
<font color="Maroon" title="c6">e1</font> = 
<font color="Maroon" title="c6">e1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="cat_8.html#K7" title="CAT_8:func.7">[x]</a> <span class="p3">(<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></span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="cat_8.html#K7" title="CAT_8:func.7">[x]</a> <span class="p3">(<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></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:117_1"><span class="lab"><font color="Green" title="E74">A7</font></span></a>, <a class="txt" href="#E2:117_1_1"><span class="lab"><font color="Green" title="E75">A8</font></span></a>, <a class="txt" href="#E4:117_1_1"><span class="lab"><font color="Green" title="E77">A10</font></span></a>, <a class="txt" href="#E5:117_1_1"><span class="lab"><font color="Green" title="E78">A11</font></span></a>, <a class="txt" href="#E1:117"><span class="lab"><font color="Green" title="E68">A1</font></span></a>, <a class="ref" href="cat_7.html#T23" onmouseover="rs('cat_7/T23')" onmouseout="rh()">CAT_7:23</a></span>
<br/>.= 
<font color="Maroon" title="c6">e1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="cat_8.html#K7" title="CAT_8:func.7">[x]</a> <span class="p2">(<span class="default"><span class="p3">(<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="p3">(<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></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:117"><span class="lab"><font color="Green" title="E71">A4</font></span></a>, <a class="txt" href="#E6:117"><span class="lab"><font color="Green" title="E73">A6</font></span></a>, <a class="txt" href="#E3:117_1_1"><span class="lab"><font color="Green" title="E76">A9</font></span></a>, <a class="ref" href="cat_8.html#T66" target="_self" onmouseover="rs('cat_8/T66')" onmouseout="rh()">Th66</a></span>
<br/>.= 
<font color="Maroon" title="c6">e1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="cat_8.html#K7" title="CAT_8:func.7">[x]</a> <span class="p2">(<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></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:117_1_1"><span class="lab"><font color="Green" title="E76">A9</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="E10:117_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A13</font></span>: 
<font color="Maroon" title="c9">g</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">h2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:117_1_1"><span class="lab"><font color="Green" title="E79">A12</font></span></a></span>;<br/>
<font color="Maroon" title="c6">e1</font> = 
<font color="Maroon" title="c6">e1</font> <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="c4">c1</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:117"><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/>.= 
<font color="Maroon" title="c6">e1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_7.html#K3" title="CAT_7:func.3">id-</a> <font color="Maroon" title="c4">c1</font></span>)</span> <a href="cat_8.html#K7" title="CAT_8:func.7">[x]</a> <span class="p2">(<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></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_8.html#T67" target="_self" onmouseover="rs('cat_8/T67')" onmouseout="rh()">Th67</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E12:117_1_1"/>
<font color="Maroon" title="c9">g</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_7.html#K3" title="CAT_7:func.3">id-</a> <font color="Maroon" title="c4">c1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:117_1_1"><span class="lab"><font color="Green" title="E79">A12</font></span></a>, <a class="txt" href="#E10:117_1_1"><span class="lab"><font color="Green" title="E80">A13</font></span></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"> <font color="Maroon" title="c8">f</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_7.html#K3" title="CAT_7:func.3">id-</a> <font color="Maroon" title="c5">c2</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c11">h3</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c5">c2</font>,<font color="Maroon" title="c5">c2</font><span class="kw"> such that </span><br/><a NAME="E14:117_1_1"/>
<font color="Maroon" title="c7">e2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">e2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">h3</font> <a href="cat_8.html#K7" title="CAT_8:func.7">[x]</a> <span class="p2">(<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></span>)</span>
 <span class="kw">and </span><br/><a NAME="E15:117_1_1"/><span class="lab"><font color="Green" title="E81">A14</font></span>: 
 for <font color="Olive" title="b1">h1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c5">c2</font>,<font color="Maroon" title="c5">c2</font>  st <font color="Maroon" title="c7">e2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">e2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">h1</font> <a href="cat_8.html#K7" title="CAT_8:func.7">[x]</a> <span class="p2">(<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></span>)</span> holds <br/><font color="Maroon" title="c11">h3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:117"><span class="lab"><font color="Green" title="E72">A5</font></span></a>, <a class="txt" href="#E2:117"><span class="lab"><font color="Green" title="E69">A2</font></span></a>, <a class="ref" href="cat_8.html#D29" target="_self" onmouseover="rs('cat_8/D29')" onmouseout="rh()">Def29</a></span>;<br/>
<font color="Maroon" title="c7">e2</font> = 
<font color="Maroon" title="c7">e2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="cat_8.html#K7" title="CAT_8:func.7">[x]</a> <span class="p3">(<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></span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="cat_8.html#K7" title="CAT_8:func.7">[x]</a> <span class="p3">(<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></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:117_1"><span class="lab"><font color="Green" title="E74">A7</font></span></a>, <a class="txt" href="#E2:117_1_1"><span class="lab"><font color="Green" title="E75">A8</font></span></a>, <a class="txt" href="#E4:117_1_1"><span class="lab"><font color="Green" title="E77">A10</font></span></a>, <a class="txt" href="#E5:117_1_1"><span class="lab"><font color="Green" title="E78">A11</font></span></a>, <a class="txt" href="#E2:117"><span class="lab"><font color="Green" title="E69">A2</font></span></a>, <a class="ref" href="cat_7.html#T23" onmouseover="rs('cat_7/T23')" onmouseout="rh()">CAT_7:23</a></span>
<br/>.= 
<font color="Maroon" title="c7">e2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c9">g</font></span>)</span> <a href="cat_8.html#K7" title="CAT_8:func.7">[x]</a> <span class="p2">(<span class="default"><span class="p3">(<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="p3">(<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></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:117"><span class="lab"><font color="Green" title="E71">A4</font></span></a>, <a class="txt" href="#E6:117"><span class="lab"><font color="Green" title="E73">A6</font></span></a>, <a class="txt" href="#E3:117_1_1"><span class="lab"><font color="Green" title="E76">A9</font></span></a>, <a class="ref" href="cat_8.html#T66" target="_self" onmouseover="rs('cat_8/T66')" onmouseout="rh()">Th66</a></span>
<br/>.= 
<font color="Maroon" title="c7">e2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c9">g</font></span>)</span> <a href="cat_8.html#K7" title="CAT_8:func.7">[x]</a> <span class="p2">(<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></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:117_1_1"><span class="lab"><font color="Green" title="E76">A9</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="E17:117_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A15</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">h3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:117_1_1"><span class="lab"><font color="Green" title="E81">A14</font></span></a></span>;<br/>
<font color="Maroon" title="c7">e2</font> = 
<font color="Maroon" title="c7">e2</font> <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="c5">c2</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:117"><span class="lab"><font color="Green" title="E69">A2</font></span></a>, <a class="ref" href="cat_7.html#T18" onmouseover="rs('cat_7/T18')" onmouseout="rh()">CAT_7:18</a></span>
<br/>.= 
<font color="Maroon" title="c7">e2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_7.html#K3" title="CAT_7:func.3">id-</a> <font color="Maroon" title="c5">c2</font></span>)</span> <a href="cat_8.html#K7" title="CAT_8:func.7">[x]</a> <span class="p2">(<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></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_8.html#T67" target="_self" onmouseover="rs('cat_8/T67')" onmouseout="rh()">Th67</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E19:117_1_1"/>
<font color="Maroon" title="c8">f</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c9">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cat_7.html#K3" title="CAT_7:func.3">id-</a> <font color="Maroon" title="c5">c2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:117_1_1"><span class="lab"><font color="Green" title="E81">A14</font></span></a>, <a class="txt" href="#E17:117_1_1"><span class="lab"><font color="Green" title="E82">A15</font></span></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/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:117_1"/>
<font color="Maroon" title="c8">f</font> is  <a href="cat_7.html#V5" title="CAT_7:attr.5">isomorphism</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:117"><span class="lab"><font color="Green" title="E71">A4</font></span></a>, <a class="txt" href="#E6:117"><span class="lab"><font color="Green" title="E73">A6</font></span></a>, <a class="ref" href="cat_7.html#D9" onmouseover="rs('cat_7/D9')" onmouseout="rh()">CAT_7:def 9</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/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E8:117"/>
<font color="Maroon" title="c4">c1</font>,<font color="Maroon" title="c5">c2</font> <a href="cat_7.html#R1" title="CAT_7:pred.1">are_isomorphic</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_7.html#D10" onmouseover="rs('cat_7/D10')" onmouseout="rh()">CAT_7:def 10</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/>


</div>
