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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_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">c1</font>, <font color="Olive" title="b2">c2</font>, <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">b</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">p1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Olive" title="b3">a</font>,<font color="Olive" title="b1">c1</font><br/>  for <font color="Olive" title="b6">p2</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Olive" title="b3">a</font>,<font color="Olive" title="b2">c2</font><br/>  for <font color="Olive" title="b7">q1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Olive" title="b4">b</font>,<font color="Olive" title="b1">c1</font><br/>  for <font color="Olive" title="b8">q2</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Olive" title="b4">b</font>,<font color="Olive" title="b2">c2</font>  st  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<font color="Olive" title="b3">a</font>,<font color="Olive" title="b1">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;  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<font color="Olive" title="b3">a</font>,<font color="Olive" title="b2">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;  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<font color="Olive" title="b4">b</font>,<font color="Olive" title="b1">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;  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<font color="Olive" title="b4">b</font>,<font color="Olive" title="b2">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; <font color="Olive" title="b3">a</font>,<font color="Olive" title="b5">p1</font>,<font color="Olive" title="b6">p2</font> <a href="cat_8.html#R1" title="CAT_8:pred.1">is_product_of</a> <font color="Olive" title="b1">c1</font>,<font color="Olive" title="b2">c2</font> &amp; <font color="Olive" title="b4">b</font>,<font color="Olive" title="b7">q1</font>,<font color="Olive" title="b8">q2</font> <a href="cat_8.html#R1" title="CAT_8:pred.1">is_product_of</a> <font color="Olive" title="b1">c1</font>,<font color="Olive" title="b2">c2</font> holds <br/><font color="Olive" title="b3">a</font>,<font color="Olive" title="b4">b</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">c1</font>, <font color="Maroon" title="c3">c2</font>, <font color="Maroon" title="c4">a</font>, <font color="Maroon" title="c5">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">  for <font color="Olive" title="b1">p1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">c1</font><br/>  for <font color="Olive" title="b2">p2</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">c2</font><br/>  for <font color="Olive" title="b3">q1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c2">c1</font><br/>  for <font color="Olive" title="b4">q2</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c3">c2</font>  st  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">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;  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">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;  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c2">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;  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c3">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; <font color="Maroon" title="c4">a</font>,<font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font> <a href="cat_8.html#R1" title="CAT_8:pred.1">is_product_of</a> <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c3">c2</font> &amp; <font color="Maroon" title="c5">b</font>,<font color="Olive" title="b3">q1</font>,<font color="Olive" title="b4">q2</font> <a href="cat_8.html#R1" title="CAT_8:pred.1">is_product_of</a> <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c3">c2</font> holds <br/><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</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">p1</font> be    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">c1</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">p2</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">c2</font><br/>  for <font color="Olive" title="b2">q1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c2">c1</font><br/>  for <font color="Olive" title="b3">q2</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c3">c2</font>  st  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">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;  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">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;  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c2">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;  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c3">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; <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">p1</font>,<font color="Olive" title="b1">p2</font> <a href="cat_8.html#R1" title="CAT_8:pred.1">is_product_of</a> <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c3">c2</font> &amp; <font color="Maroon" title="c5">b</font>,<font color="Olive" title="b2">q1</font>,<font color="Olive" title="b3">q2</font> <a href="cat_8.html#R1" title="CAT_8:pred.1">is_product_of</a> <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c3">c2</font> holds <br/><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</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">p2</font> be    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">c2</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">q1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c2">c1</font><br/>  for <font color="Olive" title="b2">q2</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c3">c2</font>  st  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">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;  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">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;  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c2">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;  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c3">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; <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font> <a href="cat_8.html#R1" title="CAT_8:pred.1">is_product_of</a> <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c3">c2</font> &amp; <font color="Maroon" title="c5">b</font>,<font color="Olive" title="b1">q1</font>,<font color="Olive" title="b2">q2</font> <a href="cat_8.html#R1" title="CAT_8:pred.1">is_product_of</a> <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c3">c2</font> holds <br/><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</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="c8">q1</font> be    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c2">c1</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">q2</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c3">c2</font>  st  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">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;  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">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;  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c2">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;  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c3">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; <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font> <a href="cat_8.html#R1" title="CAT_8:pred.1">is_product_of</a> <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c3">c2</font> &amp; <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c8">q1</font>,<font color="Olive" title="b1">q2</font> <a href="cat_8.html#R1" title="CAT_8:pred.1">is_product_of</a> <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c3">c2</font> holds <br/><font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</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="c9">q2</font> be    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c3">c2</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> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">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;  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">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;  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c2">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;  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c3">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; <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font> <a href="cat_8.html#R1" title="CAT_8:pred.1">is_product_of</a> <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c3">c2</font> &amp; <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c8">q1</font>,<font color="Maroon" title="c9">q2</font> <a href="cat_8.html#R1" title="CAT_8:pred.1">is_product_of</a> <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c3">c2</font> implies <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> <a href="cat_7.html#R1" title="CAT_7:pred.1">are_isomorphic</a>  )</span><br/>











<span class="kw">assume </span><a NAME="E1:57"/><span class="lab"><font color="Green" title="E31">A1</font></span>: 
(  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">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;  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">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;  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c2">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;  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c3">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>  )
 ; <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">a</font>,<font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font> <a href="cat_8.html#R1" title="CAT_8:pred.1">is_product_of</a> <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c3">c2</font> or  not <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c8">q1</font>,<font color="Maroon" title="c9">q2</font> <a href="cat_8.html#R1" title="CAT_8:pred.1">is_product_of</a> <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c3">c2</font> or <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> <a href="cat_7.html#R1" title="CAT_7:pred.1">are_isomorphic</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E2:57"/><span class="lab"><font color="Green" title="E32">A2</font></span>: 
<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">p1</font>,<font color="Maroon" title="c7">p2</font> <a href="cat_8.html#R1" title="CAT_8:pred.1">is_product_of</a> <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c3">c2</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">b</font>,<font color="Maroon" title="c8">q1</font>,<font color="Maroon" title="c9">q2</font> <a href="cat_8.html#R1" title="CAT_8:pred.1">is_product_of</a> <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c3">c2</font> or <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> <a href="cat_7.html#R1" title="CAT_7:pred.1">are_isomorphic</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E3:57"/><span class="lab"><font color="Green" title="E33">A3</font></span>: 
<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c8">q1</font>,<font color="Maroon" title="c9">q2</font> <a href="cat_8.html#R1" title="CAT_8:pred.1">is_product_of</a> <font color="Maroon" title="c2">c1</font>,<font color="Maroon" title="c3">c2</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">a</font>,<font color="Maroon" title="c5">b</font> <a href="cat_7.html#R1" title="CAT_7:pred.1">are_isomorphic</a> </span><br/>

<a NAME="E4:57"/>
 ex <font color="Olive" title="b1">ff</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font> ex <font color="Olive" title="b2">gg</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c4">a</font> st <br/>(  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">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> (<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c4">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; <font color="Olive" title="b2">gg</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">ff</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">a</font> &amp; <font color="Olive" title="b1">ff</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b2">gg</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">b</font> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="57_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c10">f</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</font><span class="kw"> such that </span><br/><a NAME="E2:57_1"/><span class="lab"><font color="Green" title="E34">A4</font></span>: 
( <font color="Maroon" title="c8">q1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c10">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p1</font> &amp; <font color="Maroon" title="c9">q2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c10">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p2</font> &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">a</font>,<font color="Maroon" title="c5">b</font>  st <font color="Maroon" title="c8">q1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p1</font> &amp; <font color="Maroon" title="c9">q2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p2</font> holds <br/><font color="Maroon" title="c10">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="#E1:57"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="txt" href="#E3:57"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="ref" href="cat_8.html#D10" target="_self" onmouseover="rs('cat_8/D10')" onmouseout="rh()">Def10</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">g</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c4">a</font><span class="kw"> such that </span><br/><a NAME="E4:57_1"/><span class="lab"><font color="Green" title="E35">A5</font></span>: 
( <font color="Maroon" title="c6">p1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q1</font> &amp; <font color="Maroon" title="c7">p2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c11">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q2</font> &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">b</font>,<font color="Maroon" title="c4">a</font>  st <font color="Maroon" title="c6">p1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q1</font> &amp; <font color="Maroon" title="c7">p2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q2</font> holds <br/><font color="Maroon" title="c11">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="#E1:57"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="txt" href="#E2:57"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="ref" href="cat_8.html#D10" target="_self" onmouseover="rs('cat_8/D10')" onmouseout="rh()">Def10</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c10">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">  ex <font color="Olive" title="b1">gg</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c4">a</font> st <br/>(  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">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> (<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c4">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; <font color="Olive" title="b1">gg</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c10">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">a</font> &amp; <font color="Maroon" title="c10">f</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">gg</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">b</font> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c11">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"> (  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">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> (<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c4">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; <font color="Maroon" title="c11">g</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c10">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">a</font> &amp; <font color="Maroon" title="c10">f</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c11">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">b</font> )</span><br/>

<span class="kw">thus </span><a NAME="E5:57_1"/><span class="lab"><font color="Green" title="E36">A6</font></span>: 
 <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">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="txt" href="#E1:57"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="txt" href="#E3:57"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="ref" href="cat_8.html#D10" target="_self" onmouseover="rs('cat_8/D10')" onmouseout="rh()">Def10</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"> (  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c4">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; <font color="Maroon" title="c11">g</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c10">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">a</font> &amp; <font color="Maroon" title="c10">f</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c11">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">b</font> )</span><br/>

<span class="kw">thus </span><a NAME="E6:57_1"/><span class="lab"><font color="Green" title="E37">A7</font></span>: 
 <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c4">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="txt" href="#E1:57"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="txt" href="#E2:57"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="ref" href="cat_8.html#D10" target="_self" onmouseover="rs('cat_8/D10')" onmouseout="rh()">Def10</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="c11">g</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c10">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">a</font> &amp; <font color="Maroon" title="c10">f</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c11">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">b</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c12">g11</font> = <font color="Maroon" title="c8">q1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c10">f</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">g12</font> = <font color="Maroon" title="c9">q2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c10">f</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">h1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c4">a</font><span class="kw"> such that </span><br/><a NAME="E8:57_1"/><span class="lab"><font color="Green" title="E38">A8</font></span>: 
( <font color="Maroon" title="c6">p1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c14">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c10">f</font> &amp; <font color="Maroon" title="c7">p2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c14">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c10">f</font> &amp; (  for <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">a</font>,<font color="Maroon" title="c4">a</font>  st <font color="Maroon" title="c6">p1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c10">f</font> &amp; <font color="Maroon" title="c7">p2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c10">f</font> holds <br/><font color="Maroon" title="c14">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="txt" href="#E2:57"><span class="lab"><font color="Green" title="E32">A2</font></span></a>, <a class="ref" href="cat_8.html#D10" target="_self" onmouseover="rs('cat_8/D10')" onmouseout="rh()">Def10</a></span>;<br/>
<a NAME="E9:57_1"/><span class="lab"><font color="Green" title="E39">A9</font></span>: 
<font color="Maroon" title="c6">p1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c10">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c10">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="txt" href="#E4:57_1"><span class="lab"><font color="Green" title="E35">A5</font></span></a>, <a class="txt" href="#E6:57_1"><span class="lab"><font color="Green" title="E37">A7</font></span></a>, <a class="txt" href="#E5:57_1"><span class="lab"><font color="Green" title="E36">A6</font></span></a>, <a class="ref" href="cat_7.html#T23" onmouseover="rs('cat_7/T23')" onmouseout="rh()">CAT_7:23</a></span>;<br/>
<a NAME="E10:57_1"/><span class="lab"><font color="Green" title="E40">A10</font></span>: 
<font color="Maroon" title="c7">p2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">g</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c10">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c10">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="txt" href="#E4:57_1"><span class="lab"><font color="Green" title="E35">A5</font></span></a>, <a class="txt" href="#E6:57_1"><span class="lab"><font color="Green" title="E37">A7</font></span></a>, <a class="txt" href="#E5:57_1"><span class="lab"><font color="Green" title="E36">A6</font></span></a>, <a class="ref" href="cat_7.html#T23" onmouseover="rs('cat_7/T23')" onmouseout="rh()">CAT_7:23</a></span>;<br/>
<a NAME="E11:57_1"/><span class="lab"><font color="Green" title="E41">A11</font></span>: 
<font color="Maroon" title="c6">p1</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> <font color="Maroon" title="c4">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c10">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="txt" href="#E2:57_1"><span class="lab"><font color="Green" title="E34">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/>
<a NAME="E12:57_1"/><span class="lab"><font color="Green" title="E42">A12</font></span>: 
<font color="Maroon" title="c7">p2</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> <font color="Maroon" title="c4">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c10">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="txt" href="#E2:57_1"><span class="lab"><font color="Green" title="E34">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">thus </span><font color="Maroon" title="c11">g</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c10">f</font> = 
<font color="Maroon" title="c14">h1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:57_1"><span class="lab"><font color="Green" title="E38">A8</font></span></a>, <a class="txt" href="#E9:57_1"><span class="lab"><font color="Green" title="E39">A9</font></span></a>, <a class="txt" href="#E10:57_1"><span class="lab"><font color="Green" title="E40">A10</font></span></a></span>
<br/>.= 
 <a href="cat_7.html#K3" title="CAT_7:func.3">id-</a> <font color="Maroon" title="c4">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:57_1"><span class="lab"><font color="Green" title="E38">A8</font></span></a>, <a class="txt" href="#E11:57_1"><span class="lab"><font color="Green" title="E41">A11</font></span></a>, <a class="txt" href="#E12:57_1"><span class="lab"><font color="Green" title="E42">A12</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="c10">f</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c11">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">b</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c15">g21</font> = <font color="Maroon" title="c6">p1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c11">g</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c16">g22</font> = <font color="Maroon" title="c7">p2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c11">g</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c17">h2</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c5">b</font><span class="kw"> such that </span><br/><a NAME="E15:57_1"/><span class="lab"><font color="Green" title="E43">A13</font></span>: 
( <font color="Maroon" title="c8">q1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c17">h2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c11">g</font> &amp; <font color="Maroon" title="c9">q2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c17">h2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c11">g</font> &amp; (  for <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">b</font>,<font color="Maroon" title="c5">b</font>  st <font color="Maroon" title="c8">q1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c11">g</font> &amp; <font color="Maroon" title="c9">q2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c11">g</font> holds <br/><font color="Maroon" title="c17">h2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="txt" href="#E3:57"><span class="lab"><font color="Green" title="E33">A3</font></span></a>, <a class="ref" href="cat_8.html#D10" target="_self" onmouseover="rs('cat_8/D10')" onmouseout="rh()">Def10</a></span>;<br/>
<a NAME="E16:57_1"/><span class="lab"><font color="Green" title="E44">A14</font></span>: 
<font color="Maroon" title="c8">q1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c11">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c11">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="txt" href="#E2:57_1"><span class="lab"><font color="Green" title="E34">A4</font></span></a>, <a class="txt" href="#E6:57_1"><span class="lab"><font color="Green" title="E37">A7</font></span></a>, <a class="txt" href="#E5:57_1"><span class="lab"><font color="Green" title="E36">A6</font></span></a>, <a class="ref" href="cat_7.html#T23" onmouseover="rs('cat_7/T23')" onmouseout="rh()">CAT_7:23</a></span>;<br/>
<a NAME="E17:57_1"/><span class="lab"><font color="Green" title="E45">A15</font></span>: 
<font color="Maroon" title="c9">q2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c11">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c11">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="txt" href="#E2:57_1"><span class="lab"><font color="Green" title="E34">A4</font></span></a>, <a class="txt" href="#E6:57_1"><span class="lab"><font color="Green" title="E37">A7</font></span></a>, <a class="txt" href="#E5:57_1"><span class="lab"><font color="Green" title="E36">A6</font></span></a>, <a class="ref" href="cat_7.html#T23" onmouseover="rs('cat_7/T23')" onmouseout="rh()">CAT_7:23</a></span>;<br/>
<a NAME="E18:57_1"/><span class="lab"><font color="Green" title="E46">A16</font></span>: 
<font color="Maroon" title="c8">q1</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> <font color="Maroon" title="c5">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p1</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c11">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="txt" href="#E4:57_1"><span class="lab"><font color="Green" title="E35">A5</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="E19:57_1"/><span class="lab"><font color="Green" title="E47">A17</font></span>: 
<font color="Maroon" title="c9">q2</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> <font color="Maroon" title="c5">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p2</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c11">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:57"><span class="lab"><font color="Green" title="E31">A1</font></span></a>, <a class="txt" href="#E4:57_1"><span class="lab"><font color="Green" title="E35">A5</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">thus </span><font color="Maroon" title="c10">f</font> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <font color="Maroon" title="c11">g</font> = 
<font color="Maroon" title="c17">h2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:57_1"><span class="lab"><font color="Green" title="E43">A13</font></span></a>, <a class="txt" href="#E16:57_1"><span class="lab"><font color="Green" title="E44">A14</font></span></a>, <a class="txt" href="#E17:57_1"><span class="lab"><font color="Green" title="E45">A15</font></span></a></span>
<br/>.= 
 <a href="cat_7.html#K3" title="CAT_7:func.3">id-</a> <font color="Maroon" title="c5">b</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:57_1"><span class="lab"><font color="Green" title="E43">A13</font></span></a>, <a class="txt" href="#E18:57_1"><span class="lab"><font color="Green" title="E46">A16</font></span></a>, <a class="txt" href="#E19:57_1"><span class="lab"><font color="Green" title="E47">A17</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="E5:57"/>
<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">b</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#D11" onmouseover="rs('cat_7/D11')" onmouseout="rh()">CAT_7:def 11</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>
