<?xml version="1.0"?>
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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</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> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="coh_sp.html#K28" title="COH_SP:func.28">TolCat</a> <font color="Maroon" title="c1">X</font></span>)</span><br/>  for <font color="Olive" title="b2">aa</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="coh_sp.html#K16" title="COH_SP:func.16">TOL</a> <font color="Maroon" title="c1">X</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">aa</font> holds <br/> for <font color="Olive" title="b3">i</font> being    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Olive" title="b1">a</font>,<font color="Olive" title="b1">a</font>  st <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="coh_sp.html#K23" title="COH_SP:func.23">id$</a> <font color="Olive" title="b2">aa</font> holds <br/> for <font color="Olive" title="b4">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="coh_sp.html#K28" title="COH_SP:func.28">TolCat</a> <font color="Maroon" title="c1">X</font></span>)</span> holds <br/> ( (  <a href="cat_1.html#K2" title="CAT_1:func.2">Hom</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b4">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>  implies  for <font color="Olive" title="b5">g</font> being    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Olive" title="b1">a</font>,<font color="Olive" title="b4">b</font> holds  <font color="Olive" title="b5">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">(*)</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">g</font> ) &amp; (  <a href="cat_1.html#K2" title="CAT_1:func.2">Hom</a> (<font color="Olive" title="b4">b</font>,<font color="Olive" title="b1">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>  implies  for <font color="Olive" title="b5">f</font> being    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Olive" title="b4">b</font>,<font color="Olive" title="b1">a</font> holds  <font color="Olive" title="b3">i</font> <a href="cat_1.html#K1" title="CAT_1:func.1">(*)</a> <font color="Olive" title="b5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">f</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="coh_sp.html#K28" title="COH_SP:func.28">TolCat</a> <font color="Maroon" title="c1">X</font></span>)</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">  for <font color="Olive" title="b1">aa</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="coh_sp.html#K16" title="COH_SP:func.16">TOL</a> <font color="Maroon" title="c1">X</font>  st <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">aa</font> holds <br/> for <font color="Olive" title="b2">i</font> being    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c2">a</font>  st <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="coh_sp.html#K23" title="COH_SP:func.23">id$</a> <font color="Olive" title="b1">aa</font> holds <br/> for <font color="Olive" title="b3">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="coh_sp.html#K28" title="COH_SP:func.28">TolCat</a> <font color="Maroon" title="c1">X</font></span>)</span> holds <br/> ( (  <a href="cat_1.html#K2" title="CAT_1:func.2">Hom</a> (<font color="Maroon" title="c2">a</font>,<font color="Olive" title="b3">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>  implies  for <font color="Olive" title="b4">g</font> being    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b3">b</font> holds  <font color="Olive" title="b4">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">(*)</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">g</font> ) &amp; (  <a href="cat_1.html#K2" title="CAT_1:func.2">Hom</a> (<font color="Olive" title="b3">b</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>  implies  for <font color="Olive" title="b4">f</font> being    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Olive" title="b3">b</font>,<font color="Maroon" title="c2">a</font> holds  <font color="Olive" title="b2">i</font> <a href="cat_1.html#K1" title="CAT_1:func.1">(*)</a> <font color="Olive" title="b4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">f</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">aa</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="coh_sp.html#K16" title="COH_SP:func.16">TOL</a> <font color="Maroon" title="c1">X</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="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">aa</font> implies  for <font color="Olive" title="b1">i</font> being    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c2">a</font>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="coh_sp.html#K23" title="COH_SP:func.23">id$</a> <font color="Maroon" title="c3">aa</font> holds <br/> for <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="coh_sp.html#K28" title="COH_SP:func.28">TolCat</a> <font color="Maroon" title="c1">X</font></span>)</span> holds <br/> ( (  <a href="cat_1.html#K2" title="CAT_1:func.2">Hom</a> (<font color="Maroon" title="c2">a</font>,<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>  implies  for <font color="Olive" title="b3">g</font> being    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b2">b</font> holds  <font color="Olive" title="b3">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">(*)</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> ) &amp; (  <a href="cat_1.html#K2" title="CAT_1:func.2">Hom</a> (<font color="Olive" title="b2">b</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>  implies  for <font color="Olive" title="b3">f</font> being    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Olive" title="b2">b</font>,<font color="Maroon" title="c2">a</font> holds  <font color="Olive" title="b1">i</font> <a href="cat_1.html#K1" title="CAT_1:func.1">(*)</a> <font color="Olive" title="b3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">f</font> ) ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:82"/><span class="lab"><font color="Green" title="E46">A1</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">aa</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">i</font> being    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c2">a</font>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="coh_sp.html#K23" title="COH_SP:func.23">id$</a> <font color="Maroon" title="c3">aa</font> holds <br/> for <font color="Olive" title="b2">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="coh_sp.html#K28" title="COH_SP:func.28">TolCat</a> <font color="Maroon" title="c1">X</font></span>)</span> holds <br/> ( (  <a href="cat_1.html#K2" title="CAT_1:func.2">Hom</a> (<font color="Maroon" title="c2">a</font>,<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>  implies  for <font color="Olive" title="b3">g</font> being    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b2">b</font> holds  <font color="Olive" title="b3">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">(*)</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> ) &amp; (  <a href="cat_1.html#K2" title="CAT_1:func.2">Hom</a> (<font color="Olive" title="b2">b</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>  implies  for <font color="Olive" title="b3">f</font> being    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Olive" title="b2">b</font>,<font color="Maroon" title="c2">a</font> holds  <font color="Olive" title="b1">i</font> <a href="cat_1.html#K1" title="CAT_1:func.1">(*)</a> <font color="Olive" title="b3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">f</font> ) )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">i</font> be    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c2">a</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">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="coh_sp.html#K23" title="COH_SP:func.23">id$</a> <font color="Maroon" title="c3">aa</font> implies  for <font color="Olive" title="b1">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="coh_sp.html#K28" title="COH_SP:func.28">TolCat</a> <font color="Maroon" title="c1">X</font></span>)</span> holds <br/> ( (  <a href="cat_1.html#K2" title="CAT_1:func.2">Hom</a> (<font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">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>  implies  for <font color="Olive" title="b2">g</font> being    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">b</font> holds  <font color="Olive" title="b2">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">(*)</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> ) &amp; (  <a href="cat_1.html#K2" title="CAT_1:func.2">Hom</a> (<font color="Olive" title="b1">b</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>  implies  for <font color="Olive" title="b2">f</font> being    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Olive" title="b1">b</font>,<font color="Maroon" title="c2">a</font> holds  <font color="Maroon" title="c4">i</font> <a href="cat_1.html#K1" title="CAT_1:func.1">(*)</a> <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E2:82"/><span class="lab"><font color="Green" title="E47">A2</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="coh_sp.html#K23" title="COH_SP:func.23">id$</a> <font color="Maroon" title="c3">aa</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">b</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="coh_sp.html#K28" title="COH_SP:func.28">TolCat</a> <font color="Maroon" title="c1">X</font></span>)</span> holds <br/> ( (  <a href="cat_1.html#K2" title="CAT_1:func.2">Hom</a> (<font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">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>  implies  for <font color="Olive" title="b2">g</font> being    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">b</font> holds  <font color="Olive" title="b2">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">(*)</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> ) &amp; (  <a href="cat_1.html#K2" title="CAT_1:func.2">Hom</a> (<font color="Olive" title="b1">b</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>  implies  for <font color="Olive" title="b2">f</font> being    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Olive" title="b1">b</font>,<font color="Maroon" title="c2">a</font> holds  <font color="Maroon" title="c4">i</font> <a href="cat_1.html#K1" title="CAT_1:func.1">(*)</a> <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> ) )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="coh_sp.html#K28" title="COH_SP:func.28">TolCat</a> <font color="Maroon" title="c1">X</font></span>)</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_1.html#K2" title="CAT_1:func.2">Hom</a> (<font color="Maroon" title="c2">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>  implies  for <font color="Olive" title="b1">g</font> being    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">b</font> holds  <font color="Olive" title="b1">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">(*)</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font> ) &amp; (  <a href="cat_1.html#K2" title="CAT_1:func.2">Hom</a> (<font color="Maroon" title="c5">b</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>  implies  for <font color="Olive" title="b1">f</font> being    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c2">a</font> holds  <font color="Maroon" title="c4">i</font> <a href="cat_1.html#K1" title="CAT_1:func.1">(*)</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E3:82"/>
(  <a href="cat_1.html#K2" title="CAT_1:func.2">Hom</a> (<font color="Maroon" title="c2">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>  implies  for <font color="Olive" title="b1">g</font> being    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">b</font> holds  <font color="Olive" title="b1">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">(*)</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">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_1.html#K2" title="CAT_1:func.2">Hom</a> (<font color="Maroon" title="c5">b</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>  implies  for <font color="Olive" title="b1">f</font> being    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c2">a</font> holds  <font color="Maroon" title="c4">i</font> <a href="cat_1.html#K1" title="CAT_1:func.1">(*)</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="82_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:82_1"/><span class="lab"><font color="Green" title="E48">A3</font></span>: 
 <a href="cat_1.html#K2" title="CAT_1:func.2">Hom</a> (<font color="Maroon" title="c2">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> 
 ; <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">g</font> being    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">b</font> holds  <font color="Olive" title="b1">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">(*)</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c6">g</font> be    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">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="c6">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">(*)</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">gg</font> = <font color="Maroon" title="c6">g</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="coh_sp.html#K20" title="COH_SP:func.20">MapsT</a> <font color="Maroon" title="c1">X</font> ;<br/>
<a NAME="E3:82_1"/>
 <a href="cat_1.html#K2" title="CAT_1: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/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E49">A4</font></span>:  <a href="graph_1.html#K4" title="GRAPH_1:func.4">cod</a> <font color="Maroon" title="c4">i</font> = 
<font color="Maroon" title="c2">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T5" onmouseover="rs('cat_1/T5')" onmouseout="rh()">CAT_1:5</a></span>
<br/>.= 
 <a href="graph_1.html#K3" title="GRAPH_1:func.3">dom</a> <font color="Maroon" title="c6">g</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:82_1"><span class="lab"><font color="Green" title="E48">A3</font></span></a>, <a class="ref" href="cat_1.html#T5" onmouseover="rs('cat_1/T5')" onmouseout="rh()">CAT_1:5</a></span>
;<br/>
<span class="lab"><font color="Green" title="E50">A5</font></span>:  <a href="coh_sp.html#K21" title="COH_SP:func.21">dom</a> <font color="Maroon" title="c7">gg</font> = 
 <a href="graph_1.html#K3" title="GRAPH_1:func.3">dom</a> <font color="Maroon" title="c6">g</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="coh_sp.html#D25" target="_self" onmouseover="rs('coh_sp/D25')" onmouseout="rh()">Def24</a></span>
<br/>.= 
<font color="Maroon" title="c3">aa</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:82"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E1:82_1"><span class="lab"><font color="Green" title="E48">A3</font></span></a>, <a class="ref" href="cat_1.html#T5" onmouseover="rs('cat_1/T5')" onmouseout="rh()">CAT_1:5</a></span>
;<br/>
<a NAME="E6:82_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A6</font></span>: 
 <a href="coh_sp.html#K22" title="COH_SP:func.22">cod</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K23" title="COH_SP:func.23">id$</a> <font color="Maroon" title="c3">aa</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="coh_sp.html#K21" title="COH_SP:func.21">dom</a> <font color="Maroon" title="c7">gg</font>
 
;<br/>
<a NAME="E7:82_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c4">i</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a>  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p1">(<span class="default"><a href="coh_sp.html#K28" title="COH_SP:func.28">TolCat</a> <font color="Maroon" title="c1">X</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:82_1"><span class="lab"><font color="Green" title="E49">A4</font></span></a>, <a class="ref" href="cat_1.html#D6" onmouseover="rs('cat_1/D6')" onmouseout="rh()">CAT_1:def 6</a></span>;<br/>
<span class="kw">hence </span><font color="Maroon" title="c6">g</font> <a href="cat_1.html#K1" title="CAT_1:func.1">(*)</a> <font color="Maroon" title="c4">i</font> = 
 the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p1">(<span class="default"><a href="coh_sp.html#K28" title="COH_SP:func.28">TolCat</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c4">i</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#D1" onmouseover="rs('cat_1/D1')" onmouseout="rh()">CAT_1:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c7">gg</font> <a href="coh_sp.html#K24" title="COH_SP:func.24">*</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K23" title="COH_SP:func.23">id$</a> <font color="Maroon" title="c3">aa</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:82_1"><span class="lab"><font color="Green" title="E51">A6</font></span></a>, <a class="txt" href="#E2:82"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="ref" href="coh_sp.html#D27" target="_self" onmouseover="rs('coh_sp/D27')" onmouseout="rh()">Def26</a></span>
<br/>.= 
<font color="Maroon" title="c6">g</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:82_1"><span class="lab"><font color="Green" title="E50">A5</font></span></a>, <a class="ref" href="coh_sp.html#T47" target="_self" onmouseover="rs('coh_sp/T47')" onmouseout="rh()">Th46</a></span>
;<br/> <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">thus </span><a NAME="E4:82"/>
(  <a href="cat_1.html#K2" title="CAT_1:func.2">Hom</a> (<font color="Maroon" title="c5">b</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>  implies  for <font color="Olive" title="b1">f</font> being    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c2">a</font> holds  <font color="Maroon" title="c4">i</font> <a href="cat_1.html#K1" title="CAT_1:func.1">(*)</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">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"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="82_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:82_2"/><span class="lab"><font color="Green" title="E48">A7</font></span>: 
 <a href="cat_1.html#K2" title="CAT_1:func.2">Hom</a> (<font color="Maroon" title="c5">b</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> 
 ; <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">f</font> being    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c2">a</font> holds  <font color="Maroon" title="c4">i</font> <a href="cat_1.html#K1" title="CAT_1:func.1">(*)</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c6">g</font> be    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Maroon" title="c5">b</font>,<font color="Maroon" title="c2">a</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">i</font> <a href="cat_1.html#K1" title="CAT_1:func.1">(*)</a> <font color="Maroon" title="c6">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">g</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c7">gg</font> = <font color="Maroon" title="c6">g</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="coh_sp.html#K20" title="COH_SP:func.20">MapsT</a> <font color="Maroon" title="c1">X</font> ;<br/>
<a NAME="E3:82_2"/>
 <a href="cat_1.html#K2" title="CAT_1: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/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E49">A8</font></span>:  <a href="graph_1.html#K3" title="GRAPH_1:func.3">dom</a> <font color="Maroon" title="c4">i</font> = 
<font color="Maroon" title="c2">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#T5" onmouseover="rs('cat_1/T5')" onmouseout="rh()">CAT_1:5</a></span>
<br/>.= 
 <a href="graph_1.html#K4" title="GRAPH_1:func.4">cod</a> <font color="Maroon" title="c6">g</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:82_2"><span class="lab"><font color="Green" title="E48">A7</font></span></a>, <a class="ref" href="cat_1.html#T5" onmouseover="rs('cat_1/T5')" onmouseout="rh()">CAT_1:5</a></span>
;<br/>
<span class="lab"><font color="Green" title="E50">A9</font></span>:  <a href="coh_sp.html#K22" title="COH_SP:func.22">cod</a> <font color="Maroon" title="c7">gg</font> = 
 <a href="graph_1.html#K4" title="GRAPH_1:func.4">cod</a> <font color="Maroon" title="c6">g</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="coh_sp.html#D26" target="_self" onmouseover="rs('coh_sp/D26')" onmouseout="rh()">Def25</a></span>
<br/>.= 
<font color="Maroon" title="c3">aa</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:82"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="txt" href="#E1:82_2"><span class="lab"><font color="Green" title="E48">A7</font></span></a>, <a class="ref" href="cat_1.html#T5" onmouseover="rs('cat_1/T5')" onmouseout="rh()">CAT_1:5</a></span>
;<br/>
<a NAME="E6:82_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A10</font></span>: 
 <a href="coh_sp.html#K21" title="COH_SP:func.21">dom</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K23" title="COH_SP:func.23">id$</a> <font color="Maroon" title="c3">aa</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="coh_sp.html#K22" title="COH_SP:func.22">cod</a> <font color="Maroon" title="c7">gg</font>
 
;<br/>
<a NAME="E7:82_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c6">g</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a>  the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p1">(<span class="default"><a href="coh_sp.html#K28" title="COH_SP:func.28">TolCat</a> <font color="Maroon" title="c1">X</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:82_2"><span class="lab"><font color="Green" title="E49">A8</font></span></a>, <a class="ref" href="cat_1.html#D6" onmouseover="rs('cat_1/D6')" onmouseout="rh()">CAT_1:def 6</a></span>;<br/>
<span class="kw">hence </span><font color="Maroon" title="c4">i</font> <a href="cat_1.html#K1" title="CAT_1:func.1">(*)</a> <font color="Maroon" title="c6">g</font> = 
 the <a href="cat_1.html#U1" title="CAT_1:sel.1">Comp</a> of <span class="p1">(<span class="default"><a href="coh_sp.html#K28" title="COH_SP:func.28">TolCat</a> <font color="Maroon" title="c1">X</font></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c6">g</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#D1" onmouseover="rs('cat_1/D1')" onmouseout="rh()">CAT_1:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="coh_sp.html#K23" title="COH_SP:func.23">id$</a> <font color="Maroon" title="c3">aa</font></span>)</span> <a href="coh_sp.html#K24" title="COH_SP:func.24">*</a> <font color="Maroon" title="c7">gg</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:82"><span class="lab"><font color="Green" title="E47">A2</font></span></a>, <a class="txt" href="#E6:82_2"><span class="lab"><font color="Green" title="E51">A10</font></span></a>, <a class="ref" href="coh_sp.html#D27" target="_self" onmouseover="rs('coh_sp/D27')" onmouseout="rh()">Def26</a></span>
<br/>.= 
<font color="Maroon" title="c6">g</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:82_2"><span class="lab"><font color="Green" title="E50">A9</font></span></a>, <a class="ref" href="coh_sp.html#T47" target="_self" onmouseover="rs('coh_sp/T47')" onmouseout="rh()">Th46</a></span>
;<br/> <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>


</div>
