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<span class="kw">let </span><font color="Maroon" title="c1">C</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="cat_8.html#V7" title="CAT_8:attr.7">with_binary_products</a>   <a href="cat_8.html#V8" title="CAT_8:attr.8">with_exponential_objects</a>  <a href="cat_6.html#NM2" title="CAT_6:NM.2">category</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being   <a href="cat_6.html#NM3" title="CAT_6:NM.3">Object</a> of <font color="Maroon" title="c1">C</font>  st  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<span class="p1">(<span class="default"><font color="Olive" title="b3">c</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Olive" title="b1">a</font></span>)</span>,<font color="Olive" title="b2">b</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/> ex <font color="Olive" title="b4">L</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<span class="p2">(<span class="default"><font color="Olive" title="b3">c</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Olive" title="b1">a</font></span>)</span>,<font color="Olive" title="b2">b</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<font color="Olive" title="b3">c</font>,<span class="p2">(<span class="default"><font color="Olive" title="b2">b</font> <a href="cat_8.html#K15" title="CAT_8:func.15">|^</a> <font color="Olive" title="b1">a</font></span>)</span>)</span>)</span> st <br/>( (  for <font color="Olive" title="b5">f</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Olive" title="b3">c</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font><br/>  for <font color="Olive" title="b6">h</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Olive" title="b3">c</font>,<font color="Olive" title="b2">b</font> <a href="cat_8.html#K15" title="CAT_8:func.15">|^</a> <font color="Olive" title="b1">a</font>  st <font color="Olive" title="b6">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">f</font> holds <br/><span class="p1">(<span class="default"><a href="cat_8.html#K16" title="CAT_8:func.16">eval</a> (<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>)</span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b6">h</font> <a href="cat_8.html#K7" title="CAT_8:func.7">[x]</a> <span class="p2">(<span class="default"><a href="cat_7.html#K3" title="CAT_7:func.3">id-</a> <font color="Olive" title="b1">a</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">f</font> ) &amp; <font color="Olive" title="b4">L</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">c</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"> (  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c3">b</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies  ex <font color="Olive" title="b1">L</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c3">b</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<font color="Maroon" title="c4">c</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="cat_8.html#K15" title="CAT_8:func.15">|^</a> <font color="Maroon" title="c2">a</font></span>)</span>)</span>)</span> st <br/>( (  for <font color="Olive" title="b2">f</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c4">c</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font><br/>  for <font color="Olive" title="b3">h</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</font> <a href="cat_8.html#K15" title="CAT_8:func.15">|^</a> <font color="Maroon" title="c2">a</font>  st <font color="Olive" title="b3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">f</font> holds <br/><span class="p1">(<span class="default"><a href="cat_8.html#K16" title="CAT_8:func.16">eval</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>)</span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">h</font> <a href="cat_8.html#K7" title="CAT_8:func.7">[x]</a> <span class="p2">(<span class="default"><a href="cat_7.html#K3" title="CAT_7:func.3">id-</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> ) &amp; <font color="Olive" title="b1">L</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:126"/><span class="lab"><font color="Green" title="E72">A1</font></span>: 
 <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c3">b</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">L</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c3">b</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<font color="Maroon" title="c4">c</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="cat_8.html#K15" title="CAT_8:func.15">|^</a> <font color="Maroon" title="c2">a</font></span>)</span>)</span>)</span> st <br/>( (  for <font color="Olive" title="b2">f</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c4">c</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font><br/>  for <font color="Olive" title="b3">h</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</font> <a href="cat_8.html#K15" title="CAT_8:func.15">|^</a> <font color="Maroon" title="c2">a</font>  st <font color="Olive" title="b3">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">f</font> holds <br/><span class="p1">(<span class="default"><a href="cat_8.html#K16" title="CAT_8:func.16">eval</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>)</span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">h</font> <a href="cat_8.html#K7" title="CAT_8:func.7">[x]</a> <span class="p2">(<span class="default"><a href="cat_7.html#K3" title="CAT_7:func.3">id-</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">f</font> ) &amp; <font color="Olive" title="b1">L</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  )</span><br/>

<a NAME="E2:126"/><span class="lab"><font color="Green" title="E73">A2</font></span>: 
(  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="cat_8.html#K15" title="CAT_8:func.15">|^</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c3">b</font>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c3">b</font> <a href="cat_8.html#K15" title="CAT_8:func.15">|^</a> <font color="Maroon" title="c2">a</font>, <a href="cat_8.html#K16" title="CAT_8:func.16">eval</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>) <a href="cat_8.html#R5" title="CAT_8:pred.5">is_exponent_of</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cat_8.html#T70" target="_self" onmouseover="rs('cat_8/T70')" onmouseout="rh()">Th70</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ] <span class="kw">means </span> for <font color="Olive" title="b1">f</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c4">c</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>  st <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/> ex <font color="Olive" title="b2">h</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</font> <a href="cat_8.html#K15" title="CAT_8:func.15">|^</a> <font color="Maroon" title="c2">a</font> st <br/>( <font color="Olive" title="b2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 &amp; <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cat_8.html#K16" title="CAT_8:func.16">eval</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>)</span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">h</font> <a href="cat_8.html#K7" title="CAT_8:func.7">[x]</a> <span class="p2">(<span class="default"><a href="cat_7.html#K3" title="CAT_7:func.3">id-</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> &amp; (  for <font color="Olive" title="b3">h1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</font> <a href="cat_8.html#K15" title="CAT_8:func.15">|^</a> <font color="Maroon" title="c2">a</font>  st <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cat_8.html#K16" title="CAT_8:func.16">eval</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>)</span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">h1</font> <a href="cat_8.html#K7" title="CAT_8:func.7">[x]</a> <span class="p2">(<span class="default"><a href="cat_7.html#K3" title="CAT_7:func.3">id-</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> holds <br/><font color="Olive" title="b2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">h1</font> ) );<br/>
<a NAME="E3:126"/><span class="lab"><font color="Green" title="E74">A4</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c3">b</font>) holds <br/> ex <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<font color="Maroon" title="c4">c</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="cat_8.html#K15" title="CAT_8:func.15">|^</a> <font color="Maroon" title="c2">a</font></span>)</span>) &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="126_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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"> ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c3">b</font>) implies  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<font color="Maroon" title="c4">c</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="cat_8.html#K15" title="CAT_8:func.15">|^</a> <font color="Maroon" title="c2">a</font></span>)</span>) &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">x</font>,<font color="Olive" title="b1">y</font>] ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:126_1"/><span class="lab"><font color="Green" title="E75">A5</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c3">b</font>)
 ; <a 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">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<font color="Maroon" title="c4">c</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="cat_8.html#K15" title="CAT_8:func.15">|^</a> <font color="Maroon" title="c2">a</font></span>)</span>) &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">x</font>,<font color="Olive" title="b1">y</font>] )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">f</font> = <font color="Maroon" title="c5">x</font> as    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c4">c</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:126_1"><span class="lab"><font color="Green" title="E75">A5</font></span></a>, <a class="ref" href="cat_7.html#D3" onmouseover="rs('cat_7/D3')" onmouseout="rh()">CAT_7:def 3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">y</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</font> <a href="cat_8.html#K15" title="CAT_8:func.15">|^</a> <font color="Maroon" title="c2">a</font><span class="kw"> such that </span><br/><a NAME="E4:126_1"/><span class="lab"><font color="Green" title="E76">A6</font></span>: 
( <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cat_8.html#K16" title="CAT_8:func.16">eval</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>)</span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="cat_8.html#K7" title="CAT_8:func.7">[x]</a> <span class="p2">(<span class="default"><a href="cat_7.html#K3" title="CAT_7:func.3">id-</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> &amp; (  for <font color="Olive" title="b1">h1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</font> <a href="cat_8.html#K15" title="CAT_8:func.15">|^</a> <font color="Maroon" title="c2">a</font>  st <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cat_8.html#K16" title="CAT_8:func.16">eval</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>)</span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">h1</font> <a href="cat_8.html#K7" title="CAT_8:func.7">[x]</a> <span class="p2">(<span class="default"><a href="cat_7.html#K3" title="CAT_7:func.3">id-</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> holds <br/><font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h1</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:126"><span class="lab"><font color="Green" title="E73">A2</font></span></a>, <a class="txt" href="#E1:126"><span class="lab"><font color="Green" title="E72">A1</font></span></a>, <a class="ref" href="cat_8.html#D29" target="_self" onmouseover="rs('cat_8/D29')" onmouseout="rh()">Def29</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c7">y</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="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<font color="Maroon" title="c4">c</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="cat_8.html#K15" title="CAT_8:func.15">|^</a> <font color="Maroon" title="c2">a</font></span>)</span>) &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c7">y</font>] )</span><br/>

<a NAME="E5:126_1"/>
 <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<font color="Maroon" title="c4">c</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="cat_8.html#K15" title="CAT_8:func.15">|^</a> <font color="Maroon" title="c2">a</font></span>)</span>) <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:126"><span class="lab"><font color="Green" title="E73">A2</font></span></a>, <a class="txt" href="#E1:126"><span class="lab"><font color="Green" title="E72">A1</font></span></a>, <a class="ref" href="cat_8.html#D29" target="_self" onmouseover="rs('cat_8/D29')" onmouseout="rh()">Def29</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:126_1"/>
<font color="Maroon" title="c7">y</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<font color="Maroon" title="c4">c</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="cat_8.html#K15" title="CAT_8:func.15">|^</a> <font color="Maroon" title="c2">a</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_7.html#D3" onmouseover="rs('cat_7/D3')" onmouseout="rh()">CAT_7:def 3</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">S<sub>1</sub></font>[<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c7">y</font>]</span><br/>

<span class="kw">thus </span><a NAME="E7:126_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c7">y</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:126_1"><span class="lab"><font color="Green" title="E76">A6</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">consider </span><font color="Maroon" title="c5">L</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c3">b</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<font color="Maroon" title="c4">c</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="cat_8.html#K15" title="CAT_8:func.15">|^</a> <font color="Maroon" title="c2">a</font></span>)</span>)</span>)</span><span class="kw"> such that </span><br/><a NAME="E5:126"/><span class="lab"><font color="Green" title="E75">A7</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c3">b</font>) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S1" onmouseover="rs('funct_2/S1')" onmouseout="rh()">FUNCT_2:sch 1</a>(<a class="txt" href="#E3:126"><span class="lab"><font color="Green" title="E74">A4</font></span></a>);<br/></span>
<span class="kw">take </span>
<font color="Maroon" title="c5">L</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">f</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c4">c</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font><br/>  for <font color="Olive" title="b2">h</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</font> <a href="cat_8.html#K15" title="CAT_8:func.15">|^</a> <font color="Maroon" title="c2">a</font>  st <font color="Olive" title="b2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">f</font> holds <br/><span class="p1">(<span class="default"><a href="cat_8.html#K16" title="CAT_8:func.16">eval</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>)</span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">h</font> <a href="cat_8.html#K7" title="CAT_8:func.7">[x]</a> <span class="p2">(<span class="default"><a href="cat_7.html#K3" title="CAT_7:func.3">id-</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> ) &amp; <font color="Maroon" title="c5">L</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>  )</span><br/>

<a NAME="E6:126"/><span class="lab"><font color="Green" title="E76">A8</font></span>: 
 ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<font color="Maroon" title="c4">c</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="cat_8.html#K15" title="CAT_8:func.15">|^</a> <font color="Maroon" title="c2">a</font></span>)</span>)
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="126_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c6">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E2:126_2"/><span class="lab"><font color="Green" title="E77">A9</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c3">b</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:126"><span class="lab"><font color="Green" title="E72">A1</font></span></a>, <a class="ref" href="xboole_0.html#D1" onmouseover="rs('xboole_0/D1')" onmouseout="rh()">XBOOLE_0:def 1</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E4:126_2"/><span class="lab"><font color="Green" title="E78">A10</font></span>: 
( <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<font color="Maroon" title="c4">c</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="cat_8.html#K15" title="CAT_8:func.15">|^</a> <font color="Maroon" title="c2">a</font></span>)</span>) &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font>] )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:126_2"><span class="lab"><font color="Green" title="E77">A9</font></span></a>, <a class="txt" href="#E3:126"><span class="lab"><font color="Green" title="E74">A4</font></span></a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c7">y</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="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<font color="Maroon" title="c4">c</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="cat_8.html#K15" title="CAT_8:func.15">|^</a> <font color="Maroon" title="c2">a</font></span>)</span>)</span><br/>

<span class="kw">thus </span><a NAME="E5:126_2"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<font color="Maroon" title="c4">c</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="cat_8.html#K15" title="CAT_8:func.15">|^</a> <font color="Maroon" title="c2">a</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:126_2"><span class="lab"><font color="Green" title="E78">A10</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">thus </span><a NAME="E7:126"/>
 for <font color="Olive" title="b1">f</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c4">c</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font><br/>  for <font color="Olive" title="b2">h</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</font> <a href="cat_8.html#K15" title="CAT_8:func.15">|^</a> <font color="Maroon" title="c2">a</font>  st <font color="Olive" title="b2">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">f</font> holds <br/><span class="p1">(<span class="default"><a href="cat_8.html#K16" title="CAT_8:func.16">eval</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>)</span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">h</font> <a href="cat_8.html#K7" title="CAT_8:func.7">[x]</a> <span class="p2">(<span class="default"><a href="cat_7.html#K3" title="CAT_7:func.3">id-</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <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"> <font color="Maroon" title="c5">L</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> </span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="126_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">f</font> be    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c4">c</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">h</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</font> <a href="cat_8.html#K15" title="CAT_8:func.15">|^</a> <font color="Maroon" title="c2">a</font>  st <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">f</font> holds <br/><span class="p1">(<span class="default"><a href="cat_8.html#K16" title="CAT_8:func.16">eval</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>)</span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <a href="cat_8.html#K7" title="CAT_8:func.7">[x]</a> <span class="p2">(<span class="default"><a href="cat_7.html#K3" title="CAT_7:func.3">id-</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font></span><br/>

<a NAME="E1:126_3"/>
<font color="Maroon" title="c6">f</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c3">b</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:126"><span class="lab"><font color="Green" title="E72">A1</font></span></a>, <a class="ref" href="cat_7.html#D3" onmouseover="rs('cat_7/D3')" onmouseout="rh()">CAT_7:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">h0</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</font> <a href="cat_8.html#K15" title="CAT_8:func.15">|^</a> <font color="Maroon" title="c2">a</font><span class="kw"> such that </span><br/><a NAME="E3:126_3"/><span class="lab"><font color="Green" title="E77">A11</font></span>: 
( <font color="Maroon" title="c7">h0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">f</font> &amp; <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cat_8.html#K16" title="CAT_8:func.16">eval</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>)</span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h0</font> <a href="cat_8.html#K7" title="CAT_8:func.7">[x]</a> <span class="p2">(<span class="default"><a href="cat_7.html#K3" title="CAT_7:func.3">id-</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> &amp; (  for <font color="Olive" title="b1">h1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</font> <a href="cat_8.html#K15" title="CAT_8:func.15">|^</a> <font color="Maroon" title="c2">a</font>  st <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cat_8.html#K16" title="CAT_8:func.16">eval</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>)</span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">h1</font> <a href="cat_8.html#K7" title="CAT_8:func.7">[x]</a> <span class="p2">(<span class="default"><a href="cat_7.html#K3" title="CAT_7:func.3">id-</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> holds <br/><font color="Maroon" title="c7">h0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h1</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:126"><span class="lab"><font color="Green" title="E75">A7</font></span></a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c8">h</font> be    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</font> <a href="cat_8.html#K15" title="CAT_8:func.15">|^</a> <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="c8">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">f</font> implies <span class="p1">(<span class="default"><a href="cat_8.html#K16" title="CAT_8:func.16">eval</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>)</span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">h</font> <a href="cat_8.html#K7" title="CAT_8:func.7">[x]</a> <span class="p2">(<span class="default"><a href="cat_7.html#K3" title="CAT_7:func.3">id-</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> )</span><br/>

<span class="kw">assume </span><a NAME="E4:126_3"/>
<font color="Maroon" title="c8">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">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"> <span class="p1">(<span class="default"><a href="cat_8.html#K16" title="CAT_8:func.16">eval</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>)</span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">h</font> <a href="cat_8.html#K7" title="CAT_8:func.7">[x]</a> <span class="p2">(<span class="default"><a href="cat_7.html#K3" title="CAT_7:func.3">id-</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font></span><br/>

<span class="kw">hence </span><a NAME="E5:126_3"/>
<span class="p1">(<span class="default"><a href="cat_8.html#K16" title="CAT_8:func.16">eval</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>)</span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">h</font> <a href="cat_8.html#K7" title="CAT_8:func.7">[x]</a> <span class="p2">(<span class="default"><a href="cat_7.html#K3" title="CAT_7:func.3">id-</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:126_3"><span class="lab"><font color="Green" title="E77">A11</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>

<a NAME="E8:126"/>
 for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c3">b</font>) &amp; <font color="Olive" title="b2">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c3">b</font>) &amp; <font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">x2</font> holds <br/><font color="Olive" title="b1">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">x2</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="126_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">x1</font>, <font color="Maroon" title="c7">x2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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"> ( <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c3">b</font>) &amp; <font color="Maroon" title="c7">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c3">b</font>) &amp; <font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x2</font> implies <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:126_4"/><span class="lab"><font color="Green" title="E77">A12</font></span>: 
<font color="Maroon" title="c6">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c3">b</font>)
 ; <a 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="c7">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c3">b</font>) or  not <font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x2</font> or <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> )</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">f1</font> = <font color="Maroon" title="c6">x1</font> as    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c4">c</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_7.html#D3" onmouseover="rs('cat_7/D3')" onmouseout="rh()">CAT_7:def 3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c9">h1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</font> <a href="cat_8.html#K15" title="CAT_8:func.15">|^</a> <font color="Maroon" title="c2">a</font><span class="kw"> such that </span><br/><a NAME="E4:126_4"/><span class="lab"><font color="Green" title="E78">A13</font></span>: 
( <font color="Maroon" title="c9">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x1</font> &amp; <font color="Maroon" title="c8">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cat_8.html#K16" title="CAT_8:func.16">eval</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>)</span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h1</font> <a href="cat_8.html#K7" title="CAT_8:func.7">[x]</a> <span class="p2">(<span class="default"><a href="cat_7.html#K3" title="CAT_7:func.3">id-</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> &amp; (  for <font color="Olive" title="b1">h0</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</font> <a href="cat_8.html#K15" title="CAT_8:func.15">|^</a> <font color="Maroon" title="c2">a</font>  st <font color="Maroon" title="c8">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cat_8.html#K16" title="CAT_8:func.16">eval</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>)</span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">h0</font> <a href="cat_8.html#K7" title="CAT_8:func.7">[x]</a> <span class="p2">(<span class="default"><a href="cat_7.html#K3" title="CAT_7:func.3">id-</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> holds <br/><font color="Maroon" title="c9">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h0</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:126_4"><span class="lab"><font color="Green" title="E77">A12</font></span></a>, <a class="txt" href="#E5:126"><span class="lab"><font color="Green" title="E75">A7</font></span></a></span>;<br/>
<span class="kw">assume </span><a NAME="E5:126_4"/><span class="lab"><font color="Green" title="E79">A14</font></span>: 
<font color="Maroon" title="c7">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c3">b</font>)
 ; <a 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">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x2</font> or <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font> )</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">f2</font> = <font color="Maroon" title="c7">x2</font> as    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c4">c</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_7.html#D3" onmouseover="rs('cat_7/D3')" onmouseout="rh()">CAT_7:def 3</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">h2</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</font> <a href="cat_8.html#K15" title="CAT_8:func.15">|^</a> <font color="Maroon" title="c2">a</font><span class="kw"> such that </span><br/><a NAME="E8:126_4"/><span class="lab"><font color="Green" title="E80">A15</font></span>: 
( <font color="Maroon" title="c11">h2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x2</font> &amp; <font color="Maroon" title="c10">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cat_8.html#K16" title="CAT_8:func.16">eval</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>)</span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">h2</font> <a href="cat_8.html#K7" title="CAT_8:func.7">[x]</a> <span class="p2">(<span class="default"><a href="cat_7.html#K3" title="CAT_7:func.3">id-</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> &amp; (  for <font color="Olive" title="b1">h0</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</font> <a href="cat_8.html#K15" title="CAT_8:func.15">|^</a> <font color="Maroon" title="c2">a</font>  st <font color="Maroon" title="c10">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cat_8.html#K16" title="CAT_8:func.16">eval</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>)</span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">h0</font> <a href="cat_8.html#K7" title="CAT_8:func.7">[x]</a> <span class="p2">(<span class="default"><a href="cat_7.html#K3" title="CAT_7:func.3">id-</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> holds <br/><font color="Maroon" title="c11">h2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h0</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:126_4"><span class="lab"><font color="Green" title="E79">A14</font></span></a>, <a class="txt" href="#E5:126"><span class="lab"><font color="Green" title="E75">A7</font></span></a></span>;<br/>
<span class="kw">assume </span><a NAME="E9:126_4"/>
<font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">x2</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">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font></span><br/>

<span class="kw">hence </span><a NAME="E10:126_4"/>
<font color="Maroon" title="c6">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:126_4"><span class="lab"><font color="Green" title="E78">A13</font></span></a>, <a class="txt" href="#E8:126_4"><span class="lab"><font color="Green" title="E80">A15</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E9:126"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A16</font></span>: 
<font color="Maroon" title="c5">L</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:126"><span class="lab"><font color="Green" title="E76">A8</font></span></a>, <a class="ref" href="funct_2.html#T19" onmouseover="rs('funct_2/T19')" onmouseout="rh()">FUNCT_2:19</a></span>;<br/>
<a NAME="E10:126"/>
 for <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<font color="Maroon" title="c4">c</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="cat_8.html#K15" title="CAT_8:func.15">|^</a> <font color="Maroon" title="c2">a</font></span>)</span>) holds <br/><font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c5">L</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="126_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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"> ( <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<font color="Maroon" title="c4">c</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="cat_8.html#K15" title="CAT_8:func.15">|^</a> <font color="Maroon" title="c2">a</font></span>)</span>) implies <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c5">L</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:126_5"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<font color="Maroon" title="c4">c</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="cat_8.html#K15" title="CAT_8:func.15">|^</a> <font color="Maroon" title="c2">a</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"> <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c5">L</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">h</font> = <font color="Maroon" title="c6">y</font> as    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</font> <a href="cat_8.html#K15" title="CAT_8:func.15">|^</a> <font color="Maroon" title="c2">a</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="cat_7.html#D3" onmouseover="rs('cat_7/D3')" onmouseout="rh()">CAT_7:def 3</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">f1</font> = <span class="p1">(<span class="default"><a href="cat_8.html#K16" title="CAT_8:func.16">eval</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>)</span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="cat_8.html#K7" title="CAT_8:func.7">[x]</a> <span class="p2">(<span class="default"><a href="cat_7.html#K3" title="CAT_7:func.3">id-</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span>;<br/>
<a NAME="E3:126_5"/><span class="lab"><font color="Green" title="E78">A17</font></span>: 
<span class="p1">(<span class="default"><a href="cat_8.html#K16" title="CAT_8:func.16">eval</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>)</span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="cat_8.html#K7" title="CAT_8:func.7">[x]</a> <span class="p2">(<span class="default"><a href="cat_7.html#K3" title="CAT_7:func.3">id-</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="cat_8.html#K4" title="CAT_8:func.4">[x]</a> <font color="Maroon" title="c2">a</font></span>)</span>,<font color="Maroon" title="c3">b</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:126"><span class="lab"><font color="Green" title="E72">A1</font></span></a>, <a class="ref" href="cat_7.html#D3" onmouseover="rs('cat_7/D3')" onmouseout="rh()">CAT_7:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">h1</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</font> <a href="cat_8.html#K15" title="CAT_8:func.15">|^</a> <font color="Maroon" title="c2">a</font><span class="kw"> such that </span><br/><a NAME="E5:126_5"/><span class="lab"><font color="Green" title="E79">A18</font></span>: 
( <font color="Maroon" title="c9">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_8.html#K16" title="CAT_8:func.16">eval</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>)</span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="cat_8.html#K7" title="CAT_8:func.7">[x]</a> <span class="p3">(<span class="default"><a href="cat_7.html#K3" title="CAT_7:func.3">id-</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="cat_8.html#K16" title="CAT_8:func.16">eval</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>)</span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="cat_8.html#K7" title="CAT_8:func.7">[x]</a> <span class="p2">(<span class="default"><a href="cat_7.html#K3" title="CAT_7:func.3">id-</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cat_8.html#K16" title="CAT_8:func.16">eval</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>)</span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">h1</font> <a href="cat_8.html#K7" title="CAT_8:func.7">[x]</a> <span class="p2">(<span class="default"><a href="cat_7.html#K3" title="CAT_7:func.3">id-</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> &amp; (  for <font color="Olive" title="b1">h0</font> being    <a href="cat_7.html#M1" title="CAT_7:mode.1">Morphism</a> of <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</font> <a href="cat_8.html#K15" title="CAT_8:func.15">|^</a> <font color="Maroon" title="c2">a</font>  st <span class="p1">(<span class="default"><a href="cat_8.html#K16" title="CAT_8:func.16">eval</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>)</span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="cat_8.html#K7" title="CAT_8:func.7">[x]</a> <span class="p2">(<span class="default"><a href="cat_7.html#K3" title="CAT_7:func.3">id-</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cat_8.html#K16" title="CAT_8:func.16">eval</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>)</span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">h0</font> <a href="cat_8.html#K7" title="CAT_8:func.7">[x]</a> <span class="p2">(<span class="default"><a href="cat_7.html#K3" title="CAT_7:func.3">id-</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> holds <br/><font color="Maroon" title="c9">h1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">h0</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:126"><span class="lab"><font color="Green" title="E75">A7</font></span></a></span>;<br/>
<a NAME="E6:126_5"/><span class="lab"><font color="Green" title="E80">A19</font></span>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cat_8.html#K16" title="CAT_8:func.16">eval</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>)</span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="cat_8.html#K7" title="CAT_8:func.7">[x]</a> <span class="p3">(<span class="default"><a href="cat_7.html#K3" title="CAT_7:func.3">id-</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:126_5"><span class="lab"><font color="Green" title="E79">A18</font></span></a></span>;<br/>
<a NAME="E7:126_5"/>
<span class="p1">(<span class="default"><a href="cat_8.html#K16" title="CAT_8:func.16">eval</a> (<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>)</span>)</span> <a href="cat_7.html#K4" title="CAT_7:func.4">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">h</font> <a href="cat_8.html#K7" title="CAT_8:func.7">[x]</a> <span class="p2">(<span class="default"><a href="cat_7.html#K3" title="CAT_7:func.3">id-</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:126"><span class="lab"><font color="Green" title="E76">A8</font></span></a>, <a class="txt" href="#E3:126_5"><span class="lab"><font color="Green" title="E78">A17</font></span></a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:126_5"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c5">L</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:126_5"><span class="lab"><font color="Green" title="E80">A19</font></span></a>, <a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</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>
<a NAME="E11:126"/><span class="kw">then </span>
 <a href="cat_7.html#K2" title="CAT_7:func.2">Hom</a> (<font color="Maroon" title="c4">c</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="cat_8.html#K15" title="CAT_8:func.15">|^</a> <font color="Maroon" title="c2">a</font></span>)</span>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c5">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a></span>;<br/>
<a NAME="E12:126"/><span class="kw">then </span>
<font color="Maroon" title="c5">L</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D3" onmouseover="rs('funct_2/D3')" onmouseout="rh()">FUNCT_2:def 3</a>, <a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:126"/>
<font color="Maroon" title="c5">L</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:126"><span class="lab"><font color="Green" title="E77">A16</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>
