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<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>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="cat_1.html#D10" onmouseover="rs('cat_1/D10')" onmouseout="rh()">CAT_1:def 10</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">  ex <font color="Olive">b<sub>1</sub></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 <br/> for <font color="Olive">b<sub>2</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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">b<sub>2</sub></font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  for <font color="Olive">b<sub>3</sub></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">b<sub>2</sub></font> holds  <font color="Olive">b<sub>3</sub></font> <a href="cat_1.html#K1" title="CAT_1:func.1">(*)</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>3</sub></font> ) &amp; (  <a href="cat_1.html#K2" title="CAT_1:func.2">Hom</a> (<font color="Olive">b<sub>2</sub></font>,<font color="Maroon" title="c2">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  for <font color="Olive">b<sub>3</sub></font> being    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Olive">b<sub>2</sub></font>,<font color="Maroon" title="c2">a</font> holds  <font color="Olive">b<sub>1</sub></font> <a href="cat_1.html#K1" title="CAT_1:func.1">(*)</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>3</sub></font> ) )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c3">aa</font> = <font color="Maroon" title="c2">a</font> as    <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> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">ii</font> =  <a href="coh_sp.html#K23" title="COH_SP:func.23">id$</a> <font color="Maroon" title="c3">aa</font> as   <a href="cat_1.html#NM2" title="CAT_1:NM.2">Morphism</a> of <span class="p1">(<span class="default"><a href="coh_sp.html#K28" title="COH_SP:func.28">TolCat</a> <font color="Maroon" title="c1">X</font></span>)</span> ;<br/>
<span class="lab"><font color="Green" title="E46">A1</font></span>:  <a href="graph_1.html#K4" title="GRAPH_1:func.4">cod</a> <font color="Maroon" title="c4">ii</font> = 
 <a href="coh_sp.html#K22" title="COH_SP:func.22">cod</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K23" title="COH_SP:func.23">id$</a> <font color="Maroon" title="c3">aa</font></span>)</span>
<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>

;<br/>
 <a href="graph_1.html#K3" title="GRAPH_1:func.3">dom</a> <font color="Maroon" title="c4">ii</font> = 
 <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>
<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>

;<br/>
<a NAME="E5:83_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">ii</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <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>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:83_1_1"><span class="lab"><font color="Green" title="E46">A1</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">ii</font> = <font color="Maroon" title="c4">ii</font> as    <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> <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_1.html#D5" onmouseover="rs('cat_1/D5')" onmouseout="rh()">CAT_1:def 5</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">ii</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">b<sub>1</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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">b<sub>1</sub></font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  for <font color="Olive">b<sub>2</sub></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">b<sub>1</sub></font> holds  <font color="Olive">b<sub>2</sub></font> <a href="cat_1.html#K1" title="CAT_1:func.1">(*)</a> <font color="Maroon" title="c5">ii</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> ) &amp; (  <a href="cat_1.html#K2" title="CAT_1:func.2">Hom</a> (<font color="Olive">b<sub>1</sub></font>,<font color="Maroon" title="c2">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  for <font color="Olive">b<sub>2</sub></font> being    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Maroon" title="c2">a</font> holds  <font color="Maroon" title="c5">ii</font> <a href="cat_1.html#K1" title="CAT_1:func.1">(*)</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E7:83_1_1"/>
 for <font color="Olive">b<sub>1</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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">b<sub>1</sub></font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  for <font color="Olive">b<sub>2</sub></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">b<sub>1</sub></font> holds  <font color="Olive">b<sub>2</sub></font> <a href="cat_1.html#K1" title="CAT_1:func.1">(*)</a> <font color="Maroon" title="c5">ii</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> ) &amp; (  <a href="cat_1.html#K2" title="CAT_1:func.2">Hom</a> (<font color="Olive">b<sub>1</sub></font>,<font color="Maroon" title="c2">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  for <font color="Olive">b<sub>2</sub></font> being    <a href="cat_1.html#M1" title="CAT_1:mode.1">Morphism</a> of <font color="Olive">b<sub>1</sub></font>,<font color="Maroon" title="c2">a</font> holds  <font color="Maroon" title="c5">ii</font> <a href="cat_1.html#K1" title="CAT_1:func.1">(*)</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="coh_sp.html#E81"><span class="lab"><font color="Green" title="E45">Lm5</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/>


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