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<a NAME="E1:31_1"/><i><font color="Green" title="E18">A9</font></i>: 
 ex <font color="Olive" title="b1">F</font> being <a href="functor0.html#V15" title="FUNCTOR0:attr.15">covariant</a>  <a href="functor0.html#M2" title="FUNCTOR0:mode.2">Functor</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() st <br/>( ( for <font color="Olive" title="b2">a</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>1</sub></font>() holds  <font color="Olive" title="b1">F</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b2">a</font>) ) &amp; ( for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>1</sub></font>()  st <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></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>  holds <br/>for <font color="Olive" title="b4">f</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font> holds  <font color="Olive" title="b1">F</font> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Olive" title="b4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font>,<font color="Olive" title="b4">f</font>) ) )
 
<b>by </b><i><a class="txt" href="yellow21.html#E3:31"><i><font color="Green" title="E12">A3</font></i></a></i>;<br/>
<a NAME="E2:31_1"/><i><font color="Green" title="E19">A10</font></i>: 
 ex <font color="Olive" title="b1">G</font> being <a href="functor0.html#V15" title="FUNCTOR0:attr.15">covariant</a>  <a href="functor0.html#M2" title="FUNCTOR0:mode.2">Functor</a> of <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>1</sub></font>() st <br/>( ( for <font color="Olive" title="b2">a</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>2</sub></font>() holds  <font color="Olive" title="b1">G</font> <a href="functor0.html#K3" title="FUNCTOR0:func.3">.</a> <font color="Olive" title="b2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b2">a</font>) ) &amp; ( for <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>2</sub></font>()  st <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></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>  holds <br/>for <font color="Olive" title="b4">f</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font> holds  <font color="Olive" title="b1">G</font> <a href="functor0.html#K6" title="FUNCTOR0:func.6">.</a> <font color="Olive" title="b4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font>,<font color="Olive" title="b4">f</font>) ) )
 
<b>by </b><i><a class="txt" href="yellow21.html#E4:31"><i><font color="Green" title="E13">A4</font></i></a></i>;<br/>
<a NAME="E3:31_1"/><i><font color="Green" title="E20">A11</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b2">b</font>)) holds <br/>( <font color="Maroon">F<sub>7</sub></font>(<font color="Olive" title="b2">b</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <font color="Maroon">F<sub>7</sub></font>(<font color="Olive" title="b2">b</font>) <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b2">b</font>,<font color="Olive" title="b1">a</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <font color="Maroon">F<sub>7</sub></font>(<font color="Olive" title="b2">b</font>) is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>1</sub></font>(); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c2">b</font>)) implies ( <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c2">b</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c2">b</font>) <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c1">a</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c2">b</font>) is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> ) )</span><br/>

<b>assume </b><a NAME="E1:31_1_1"/><i><font color="Green" title="E21">A12</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c2">b</font>))
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c2">b</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c2">b</font>) <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c1">a</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c2">b</font>) is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> )</span><br/>

<a NAME="E2:31_1_1"/><i><font color="Green" title="E22">A13</font></i>: 
(  <a href="yellow21.html#K3" title="YELLOW21:func.3">latt</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> &amp;  <a href="yellow21.html#K3" title="YELLOW21:func.3">latt</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> )
 
;<br/>
<b>consider </b><font color="Maroon" title="c3">f</font> being  <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c2">b</font>)),<span class="p1">(<span class="default"><a href="yellow21.html#K3" title="YELLOW21:func.3">latt</a> <font color="Maroon" title="c2">b</font></span>)</span><b> such that </b><br/><a NAME="E4:31_1_1"/><i><font color="Green" title="E23">A14</font></i>: 
( <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c2">b</font>) &amp; <font color="Maroon" title="c3">f</font> is <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c2">b</font>)), <a href="yellow21.html#K3" title="YELLOW21:func.3">latt</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">f</font>] &amp; <font color="Maroon">P<sub>1</sub></font>[ <a href="yellow21.html#K3" title="YELLOW21:func.3">latt</a> <font color="Maroon" title="c2">b</font>,<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c2">b</font>)),<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> ] )
 <b>by </b><i><a class="txt" href="yellow21.html#E5:31"><i><font color="Green" title="E14">A5</font></i></a></i>;<br/>
<b>thus </b><a NAME="E5:31_1_1"/>
<font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c2">b</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 <b>by </b><i><a class="txt" href="yellow21.html#E1:31"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="yellow21.html#E1:31_1_1"><i><font color="Green" title="E21">A12</font></i></a>, <a class="txt" href="yellow21.html#E2:31_1_1"><i><font color="Green" title="E22">A13</font></i></a>, <a class="txt" href="yellow21.html#E4:31_1_1"><i><font color="Green" title="E23">A14</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c2">b</font>) <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c1">a</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c2">b</font>) is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> )</span><br/>

<a NAME="E6:31_1_1"/>
 ex <font color="Olive" title="b1">g</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="yellow21.html#K3" title="YELLOW21:func.3">latt</a> <font color="Maroon" title="c2">b</font></span>)</span>,<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c2">b</font>)) st <br/>( <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  &amp; <font color="Olive" title="b1">g</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> )
 
<b>by </b><i><a class="txt" href="yellow21.html#E4:31_1_1"><i><font color="Green" title="E23">A14</font></i></a>, <a class="ref" href="waybel_0.html#D38" title="WAYBEL_0:def.38">WAYBEL_0:def 38</a></i>;<br/>
<b>hence </b><a NAME="E7:31_1_1"/>
<font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c2">b</font>) <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c1">a</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 <b>by </b><i><a class="txt" href="yellow21.html#E1:31"><i><font color="Green" title="E10">A1</font></i></a>, <a class="txt" href="yellow21.html#E1:31_1_1"><i><font color="Green" title="E21">A12</font></i></a>, <a class="txt" href="yellow21.html#E2:31_1_1"><i><font color="Green" title="E22">A13</font></i></a>, <a class="txt" href="yellow21.html#E4:31_1_1"><i><font color="Green" title="E23">A14</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c2">b</font>) is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a></span><br/>

<b>thus </b><a NAME="E8:31_1_1"/>
<font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c2">b</font>) is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="txt" href="yellow21.html#E4:31_1_1"><i><font color="Green" title="E23">A14</font></i></a>, <a class="ref" href="waybel_0.html#T66" title="WAYBEL_0:th.66">WAYBEL_0:66</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E4:31_1"/><i><font color="Green" title="E21">A15</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>2</sub></font>()  st <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">a</font>)) holds <br/>( <font color="Maroon">F<sub>8</sub></font>(<font color="Olive" title="b1">a</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <font color="Maroon">F<sub>8</sub></font>(<font color="Olive" title="b1">a</font>) <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b2">b</font>,<font color="Olive" title="b1">a</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <font color="Maroon">F<sub>8</sub></font>(<font color="Olive" title="b1">a</font>) is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>2</sub></font>(); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c1">a</font>)) implies ( <font color="Maroon">F<sub>8</sub></font>(<font color="Maroon" title="c1">a</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <font color="Maroon">F<sub>8</sub></font>(<font color="Maroon" title="c1">a</font>) <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c1">a</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <font color="Maroon">F<sub>8</sub></font>(<font color="Maroon" title="c1">a</font>) is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> ) )</span><br/>

<b>assume </b><a NAME="E1:31_1_2"/><i><font color="Green" title="E22">A16</font></i>: 
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c1">a</font>))
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">F<sub>8</sub></font>(<font color="Maroon" title="c1">a</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <font color="Maroon">F<sub>8</sub></font>(<font color="Maroon" title="c1">a</font>) <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c1">a</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <font color="Maroon">F<sub>8</sub></font>(<font color="Maroon" title="c1">a</font>) is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> )</span><br/>

<a NAME="E2:31_1_2"/><i><font color="Green" title="E23">A17</font></i>: 
(  <a href="yellow21.html#K3" title="YELLOW21:func.3">latt</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> &amp;  <a href="yellow21.html#K3" title="YELLOW21:func.3">latt</a> <font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> )
 
;<br/>
<b>consider </b><font color="Maroon" title="c3">f</font> being  <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="yellow21.html#K3" title="YELLOW21:func.3">latt</a> <font color="Maroon" title="c1">a</font></span>)</span>,<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c1">a</font>))<b> such that </b><br/><a NAME="E4:31_1_2"/><i><font color="Green" title="E24">A18</font></i>: 
( <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>8</sub></font>(<font color="Maroon" title="c1">a</font>) &amp; <font color="Maroon" title="c3">f</font> is <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a> &amp; <font color="Maroon">P<sub>2</sub></font>[ <a href="yellow21.html#K3" title="YELLOW21:func.3">latt</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c1">a</font>)),<font color="Maroon" title="c3">f</font>] &amp; <font color="Maroon">P<sub>2</sub></font>[<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c1">a</font>)), <a href="yellow21.html#K3" title="YELLOW21:func.3">latt</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> ] )
 <b>by </b><i><a class="txt" href="yellow21.html#E6:31"><i><font color="Green" title="E15">A6</font></i></a></i>;<br/>
<b>thus </b><a NAME="E5:31_1_2"/>
<font color="Maroon">F<sub>8</sub></font>(<font color="Maroon" title="c1">a</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 <b>by </b><i><a class="txt" href="yellow21.html#E2:31"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="yellow21.html#E1:31_1_2"><i><font color="Green" title="E22">A16</font></i></a>, <a class="txt" href="yellow21.html#E2:31_1_2"><i><font color="Green" title="E23">A17</font></i></a>, <a class="txt" href="yellow21.html#E4:31_1_2"><i><font color="Green" title="E24">A18</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">F<sub>8</sub></font>(<font color="Maroon" title="c1">a</font>) <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c1">a</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> &amp; <font color="Maroon">F<sub>8</sub></font>(<font color="Maroon" title="c1">a</font>) is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> )</span><br/>

<a NAME="E6:31_1_2"/>
 ex <font color="Olive" title="b1">g</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c1">a</font>)),<span class="p1">(<span class="default"><a href="yellow21.html#K3" title="YELLOW21:func.3">latt</a> <font color="Maroon" title="c1">a</font></span>)</span> st <br/>( <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  &amp; <font color="Olive" title="b1">g</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> )
 
<b>by </b><i><a class="txt" href="yellow21.html#E4:31_1_2"><i><font color="Green" title="E24">A18</font></i></a>, <a class="ref" href="waybel_0.html#D38" title="WAYBEL_0:def.38">WAYBEL_0:def 38</a></i>;<br/>
<b>hence </b><a NAME="E7:31_1_2"/>
<font color="Maroon">F<sub>8</sub></font>(<font color="Maroon" title="c1">a</font>) <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c1">a</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span>
 <b>by </b><i><a class="txt" href="yellow21.html#E2:31"><i><font color="Green" title="E11">A2</font></i></a>, <a class="txt" href="yellow21.html#E1:31_1_2"><i><font color="Green" title="E22">A16</font></i></a>, <a class="txt" href="yellow21.html#E2:31_1_2"><i><font color="Green" title="E23">A17</font></i></a>, <a class="txt" href="yellow21.html#E4:31_1_2"><i><font color="Green" title="E24">A18</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">F<sub>8</sub></font>(<font color="Maroon" title="c1">a</font>) is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a></span><br/>

<b>thus </b><a NAME="E8:31_1_2"/>
<font color="Maroon">F<sub>8</sub></font>(<font color="Maroon" title="c1">a</font>) is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="txt" href="yellow21.html#E4:31_1_2"><i><font color="Green" title="E24">A18</font></i></a>, <a class="ref" href="waybel_0.html#T66" title="WAYBEL_0:th.66">WAYBEL_0:66</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E5:31_1"/><i><font color="Green" title="E22">A19</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>1</sub></font>()  st <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></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>  holds <br/>for <font color="Olive" title="b3">f</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> holds  <font color="Maroon">F<sub>7</sub></font>(<font color="Olive" title="b2">b</font>) <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b1">a</font>),<font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b2">b</font>),<font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">f</font>)) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Olive" title="b1">a</font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>1</sub></font>(); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></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>  implies for <font color="Olive" title="b1">f</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> holds  <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c2">b</font>) <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c1">a</font>),<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c2">b</font>),<font color="Maroon">F<sub>5</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<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 href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c1">a</font>) )</span><br/>

<b>assume </b><a NAME="E1:31_1_3"/><i><font color="Green" title="E23">A20</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> holds  <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c2">b</font>) <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c1">a</font>),<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c2">b</font>),<font color="Maroon">F<sub>5</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<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 href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c1">a</font>)</span><br/>

<b>let </b><font color="Maroon" title="c3">f</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c2">b</font>) <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c1">a</font>),<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c2">b</font>),<font color="Maroon">F<sub>5</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">f</font>)) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c1">a</font>)</span><br/>

<a NAME="E2:31_1_3"/>
 <a href="yellow21.html#K5" title="YELLOW21:func.5">@</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font>
 
<b>by </b><i><a class="txt" href="yellow21.html#E1:31_1_3"><i><font color="Green" title="E23">A20</font></i></a>, <a class="ref" href="yellow21.html#D7" target="_self" title="YELLOW21:def.7">Def7</a></i>;<br/>
<b>hence </b><a NAME="E3:31_1_3"/>
<font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c2">b</font>) <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon">F<sub>6</sub></font>(<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c1">a</font>),<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c2">b</font>),<font color="Maroon">F<sub>5</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">f</font>)) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon">F<sub>7</sub></font>(<font color="Maroon" title="c1">a</font>)
 <b>by </b><i><a class="txt" href="yellow21.html#E7:31"><i><font color="Green" title="E16">A7</font></i></a>, <a class="txt" href="yellow21.html#E1:31_1_3"><i><font color="Green" title="E23">A20</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E6:31_1"/><i><font color="Green" title="E23">A21</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>2</sub></font>()  st <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></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>  holds <br/>for <font color="Olive" title="b3">f</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> holds  <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">a</font>),<font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b2">b</font>),<font color="Maroon">F<sub>6</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">f</font>)) <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon">F<sub>8</sub></font>(<font color="Olive" title="b1">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>8</sub></font>(<font color="Olive" title="b2">b</font>) <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b3">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon">F<sub>2</sub></font>(); <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></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>  implies for <font color="Olive" title="b1">f</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> holds  <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c1">a</font>),<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c2">b</font>),<font color="Maroon">F<sub>6</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Olive" title="b1">f</font>)) <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon">F<sub>8</sub></font>(<font color="Maroon" title="c1">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>8</sub></font>(<font color="Maroon" title="c2">b</font>) <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b1">f</font> )</span><br/>

<b>assume </b><a NAME="E1:31_1_4"/><i><font color="Green" title="E24">A22</font></i>: 
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> holds  <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c1">a</font>),<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c2">b</font>),<font color="Maroon">F<sub>6</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Olive" title="b1">f</font>)) <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon">F<sub>8</sub></font>(<font color="Maroon" title="c1">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>8</sub></font>(<font color="Maroon" title="c2">b</font>) <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b1">f</font></span><br/>

<b>let </b><font color="Maroon" title="c3">f</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c1">a</font>),<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c2">b</font>),<font color="Maroon">F<sub>6</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">f</font>)) <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon">F<sub>8</sub></font>(<font color="Maroon" title="c1">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>8</sub></font>(<font color="Maroon" title="c2">b</font>) <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">f</font></span><br/>

<a NAME="E2:31_1_4"/>
 <a href="yellow21.html#K5" title="YELLOW21:func.5">@</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font>
 
<b>by </b><i><a class="txt" href="yellow21.html#E1:31_1_4"><i><font color="Green" title="E24">A22</font></i></a>, <a class="ref" href="yellow21.html#D7" target="_self" title="YELLOW21:def.7">Def7</a></i>;<br/>
<b>hence </b><a NAME="E3:31_1_4"/>
<font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c1">a</font>),<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c2">b</font>),<font color="Maroon">F<sub>6</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">f</font>)) <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon">F<sub>8</sub></font>(<font color="Maroon" title="c1">a</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>8</sub></font>(<font color="Maroon" title="c2">b</font>) <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c3">f</font>
 <b>by </b><i><a class="txt" href="yellow21.html#E8:31"><i><font color="Green" title="E17">A8</font></i></a>, <a class="txt" href="yellow21.html#E1:31_1_4"><i><font color="Green" title="E24">A22</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E7:31_1"/>
<font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() <a href="yellow18.html#R1" title="YELLOW18:pred.1">are_equivalent</a> 
 <b>from </b><i><a class="ref" href="yellow18.html#S22" title="YELLOW18:sch.22">YELLOW18:sch 22</a>(<a class="txt" href="yellow21.html#E1:31_1"><i><font color="Green" title="E18">A9</font></i></a>, <a class="txt" href="yellow21.html#E2:31_1"><i><font color="Green" title="E19">A10</font></i></a>, <a class="txt" href="yellow21.html#E3:31_1"><i><font color="Green" title="E20">A11</font></i></a>, <a class="txt" href="yellow21.html#E4:31_1"><i><font color="Green" title="E21">A15</font></i></a>, <a class="txt" href="yellow21.html#E5:31_1"><i><font color="Green" title="E22">A19</font></i></a>, <a class="txt" href="yellow21.html#E6:31_1"><i><font color="Green" title="E23">A21</font></i></a>);</i> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
