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<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = $4 <a href="zf_fund1.html#K1" title="ZF_FUND1:func.1">(#)</a> $5;<br/>
<div><i><font color="Green" title="E40">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1"><b>now </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="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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"> for <font color="Olive" title="b1">f</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) holds <br/><font color="Olive" title="b1">f</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a></span><br/><b>let </b><font color="Maroon" title="c3">f</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) implies <font color="Maroon" title="c3">f</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> )</span><br/><b>assume </b><a NAME="E1:75_1_1"/><i><font color="Green" title="E41">A5</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>(<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" title="c3">f</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a></span><br/><a NAME="E2:75_1_1"/>
<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <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>)
 <b>by </b><i><a class="txt" href="yellow18.html#E2:75"><i><font color="Green" title="E38">A2</font></i></a></i>;<br/><a NAME="E3:75_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">f'</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> st <br/>( <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f'</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Olive" title="b1">f'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c1">a</font>) &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">f'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c2">b</font>) )
 <b>by </b><i><a class="txt" href="yellow18.html#E1:75_1_1"><i><font color="Green" title="E41">A5</font></i></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/><b>hence </b><a NAME="E4:75_1_1"/>
<font color="Maroon" title="c3">f</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>
 ; <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="E2:75_1"/><i><font color="Green" title="E41">A6</font></i>: 
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="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive" title="b4">f</font>, <font color="Olive" title="b5">g</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>) &amp; <font color="Olive" title="b5">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font>) holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">f</font>,<font color="Olive" title="b5">g</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</font>)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_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>, <font color="Maroon" title="c3">c</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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"> for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) &amp; <font color="Olive" title="b2">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>) holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">g</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font>)</span><br/><b>let </b><font color="Maroon" title="c4">f</font>, <font color="Maroon" title="c5">g</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) &amp; <font color="Maroon" title="c5">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>) implies <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font>) )</span><br/>



<b>assume </b><a NAME="E1:75_1_2"/><i><font color="Green" title="E42">A7</font></i>: 
( <font color="Maroon" title="c4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) &amp; <font color="Maroon" title="c5">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font>)</span><br/>

<b>then reconsider </b><font color="Maroon" title="c6">f</font> = <font color="Maroon" title="c4">f</font>, <font color="Maroon" title="c7">g</font> = <font color="Maroon" title="c5">g</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <b>by </b><i><a class="txt" href="yellow18.html#E1:75_1"><i><font color="Green" title="E40">A4</font></i></a></i>;<br/>
<a NAME="E3:75_1_2"/>
<font color="Maroon" title="c7">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="zf_fund1.html#K1" title="ZF_FUND1:func.1">(#)</a> <font color="Maroon" title="c7">g</font>
 
<b>by </b><i><a class="ref" href="yellow16.html#T1" title="YELLOW16:th.1">YELLOW16:1</a></i>;<br/>
<b>hence </b><a NAME="E4:75_1_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">g</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font>)
 <b>by </b><i><a class="txt" href="yellow18.html#E1:75"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="yellow18.html#E1:75_1_2"><i><font color="Green" title="E42">A7</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="E3:75_1"/><i><font color="Green" title="E42">A8</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font>, <font color="Olive" title="b4">d</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive" title="b5">f</font>, <font color="Olive" title="b6">g</font>, <font color="Olive" title="b7">h</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b5">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>) &amp; <font color="Olive" title="b6">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font>) &amp; <font color="Olive" title="b7">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">d</font>) holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">d</font>,<font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font>,<font color="Olive" title="b5">f</font>,<font color="Olive" title="b6">g</font>),<font color="Olive" title="b7">h</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b4">d</font>,<font color="Olive" title="b5">f</font>,<font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">d</font>,<font color="Olive" title="b6">g</font>,<font color="Olive" title="b7">h</font>))
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_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>, <font color="Maroon" title="c3">c</font>, <font color="Maroon" title="c4">d</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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"> for <font color="Olive" title="b1">f</font>, <font color="Olive" title="b2">g</font>, <font color="Olive" title="b3">h</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) &amp; <font color="Olive" title="b2">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>) &amp; <font color="Olive" title="b3">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>) holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Olive" title="b1">f</font>,<font color="Olive" title="b2">g</font>),<font color="Olive" title="b3">h</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">d</font>,<font color="Olive" title="b1">f</font>,<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>,<font color="Olive" title="b2">g</font>,<font color="Olive" title="b3">h</font>))</span><br/><b>let </b><font color="Maroon" title="c5">f</font>, <font color="Maroon" title="c6">g</font>, <font color="Maroon" title="c7">h</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) &amp; <font color="Maroon" title="c6">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>) &amp; <font color="Maroon" title="c7">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>) implies <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font>),<font color="Maroon" title="c7">h</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">f</font>,<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c7">h</font>)) )</span><br/>



<b>assume </b><a NAME="E1:75_1_3"/>
( <font color="Maroon" title="c5">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>) &amp; <font color="Maroon" title="c6">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>) &amp; <font color="Maroon" title="c7">h</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font>),<font color="Maroon" title="c7">h</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">f</font>,<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c7">h</font>))</span><br/>

<b>then reconsider </b><font color="Maroon" title="c8">f</font> = <font color="Maroon" title="c5">f</font>, <font color="Maroon" title="c9">g</font> = <font color="Maroon" title="c6">g</font>, <font color="Maroon" title="c10">h</font> = <font color="Maroon" title="c7">h</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> <b>by </b><i><a class="txt" href="yellow18.html#E1:75_1"><i><font color="Green" title="E40">A4</font></i></a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="zf_fund1.html#K1" title="ZF_FUND1:func.1">(#)</a> <font color="Maroon" title="c9">g</font></span>)</span> <a href="zf_fund1.html#K1" title="ZF_FUND1:func.1">(#)</a> <font color="Maroon" title="c10">h</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="zf_fund1.html#K1" title="ZF_FUND1:func.1">(#)</a> <font color="Maroon" title="c10">h</font>
<b>by </b><i><a class="ref" href="yellow16.html#T1" title="YELLOW16:th.1">YELLOW16:1</a></i>
<br/>.= 
<font color="Maroon" title="c10">h</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">g</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c8">f</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow16.html#T1" title="YELLOW16:th.1">YELLOW16:1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">h</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c9">g</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c8">f</font>
<b>by </b><i><a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a></i>
<br/>.= 
<font color="Maroon" title="c8">f</font> <a href="zf_fund1.html#K1" title="ZF_FUND1:func.1">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">h</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c9">g</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow16.html#T1" title="YELLOW16:th.1">YELLOW16:1</a></i>
;<br/>
<b>hence </b><a NAME="E4:75_1_3"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font>),<font color="Maroon" title="c7">h</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">f</font>,<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c6">g</font>,<font color="Maroon" title="c7">h</font>))
 <b>by </b><i><a class="ref" href="yellow16.html#T1" title="YELLOW16:th.1">YELLOW16:1</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="E4:75_1"/><i><font color="Green" title="E43">A9</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()  ex <font color="Olive" title="b2">f</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b1">a</font>) &amp; ( for <font color="Olive" title="b3">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive" title="b4">g</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b4">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">b</font>) holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">b</font>,<font color="Olive" title="b2">f</font>,<font color="Olive" title="b4">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">g</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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">  ex <font color="Olive" title="b1">f</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c1">a</font>) &amp; ( for <font color="Olive" title="b2">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive" title="b3">g</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Olive" title="b2">b</font>) holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b1">f</font>,<font color="Olive" title="b3">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> ) )</span><br/>

<b>take </b><font color="Maroon" title="c2">f</font> =  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon">F<sub>3</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" title="c2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c1">a</font>) &amp; ( for <font color="Olive" title="b1">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive" title="b2">g</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Olive" title="b1">b</font>) holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c1">a</font>,<font color="Olive" title="b1">b</font>,<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b2">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> ) )</span><br/>

<b>thus </b><a NAME="E1:75_1_4"/>
<font color="Maroon" title="c2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c1">a</font>)
 <b>by </b><i><a class="txt" href="yellow18.html#E3:75"><i><font color="Green" title="E39">A3</font></i></a></i>; <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">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive" title="b2">g</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Olive" title="b1">b</font>) holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c1">a</font>,<font color="Olive" title="b1">b</font>,<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b2">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font></span><br/>

<b>let </b><font color="Maroon" title="c3">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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"> for <font color="Olive" title="b1">g</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">b</font>) holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">f</font>,<font color="Olive" title="b1">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font></span><br/><b>let </b><font color="Maroon" title="c4">g</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">b</font>) implies <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> )</span><br/>



<b>assume </b><a NAME="E2:75_1_4"/><i><font color="Green" title="E44">A10</font></i>: 
<font color="Maroon" title="c4">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">b</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c4">g</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font></span><br/>

<a NAME="E3:75_1_4"/>
<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c3">b</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <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="c3">b</font>)
 
<b>by </b><i><a class="txt" href="yellow18.html#E2:75"><i><font color="Green" title="E38">A2</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">h</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E5:75_1_4"/><i><font color="Green" title="E45">A11</font></i>: 
( <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c1">a</font>) &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c3">b</font>) )
 <b>by </b><i><a class="txt" href="yellow18.html#E2:75_1_4"><i><font color="Green" title="E44">A10</font></i></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c2">f</font> <a href="zf_fund1.html#K1" title="ZF_FUND1:func.1">(#)</a> <font color="Maroon" title="c4">g</font> = 
<font color="Maroon" title="c5">h</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c2">f</font>
<b>by </b><i><a class="txt" href="yellow18.html#E5:75_1_4"><i><font color="Green" title="E45">A11</font></i></a>, <a class="ref" href="yellow16.html#T1" title="YELLOW16:th.1">YELLOW16:1</a></i>
<br/>.= 
<font color="Maroon" title="c4">g</font>
<b>by </b><i><a class="txt" href="yellow18.html#E5:75_1_4"><i><font color="Green" title="E45">A11</font></i></a>, <a class="ref" href="relat_1.html#T78" title="RELAT_1:th.78">RELAT_1:78</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="E5:75_1"/><i><font color="Green" title="E44">A12</font></i>: 
for <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()  ex <font color="Olive" title="b2">f</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b1">a</font>) &amp; ( for <font color="Olive" title="b3">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive" title="b4">g</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b4">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b3">b</font>,<font color="Olive" title="b1">a</font>) holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b3">b</font>,<font color="Olive" title="b1">a</font>,<font color="Olive" title="b1">a</font>,<font color="Olive" title="b4">g</font>,<font color="Olive" title="b2">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">g</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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">  ex <font color="Olive" title="b1">f</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c1">a</font>) &amp; ( for <font color="Olive" title="b2">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive" title="b3">g</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b3">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b2">b</font>,<font color="Maroon" title="c1">a</font>) holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b2">b</font>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c1">a</font>,<font color="Olive" title="b3">g</font>,<font color="Olive" title="b1">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">g</font> ) )</span><br/>

<b>take </b><font color="Maroon" title="c2">f</font> =  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon">F<sub>3</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" title="c2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c1">a</font>) &amp; ( for <font color="Olive" title="b1">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive" title="b2">g</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">b</font>,<font color="Maroon" title="c1">a</font>) holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">b</font>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c1">a</font>,<font color="Olive" title="b2">g</font>,<font color="Maroon" title="c2">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font> ) )</span><br/>

<b>thus </b><a NAME="E1:75_1_5"/>
<font color="Maroon" title="c2">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c1">a</font>)
 <b>by </b><i><a class="txt" href="yellow18.html#E3:75"><i><font color="Green" title="E39">A3</font></i></a></i>; <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">b</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive" title="b2">g</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">b</font>,<font color="Maroon" title="c1">a</font>) holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">b</font>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c1">a</font>,<font color="Olive" title="b2">g</font>,<font color="Maroon" title="c2">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">g</font></span><br/>

<b>let </b><font color="Maroon" title="c3">b</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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"> for <font color="Olive" title="b1">g</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c1">a</font>) holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c1">a</font>,<font color="Olive" title="b1">g</font>,<font color="Maroon" title="c2">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">g</font></span><br/><b>let </b><font color="Maroon" title="c4">g</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c1">a</font>) implies <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c2">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font> )</span><br/>



<b>assume </b><a NAME="E2:75_1_5"/><i><font color="Green" title="E45">A13</font></i>: 
<font color="Maroon" title="c4">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c3">b</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">H<sub>1</sub></font>(<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c4">g</font>,<font color="Maroon" title="c2">f</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font></span><br/>

<a NAME="E3:75_1_5"/>
<font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c1">a</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c3">b</font>),<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c1">a</font>)
 
<b>by </b><i><a class="txt" href="yellow18.html#E2:75"><i><font color="Green" title="E38">A2</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">h</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E5:75_1_5"/><i><font color="Green" title="E46">A14</font></i>: 
( <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">h</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c3">b</font>) &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c5">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c1">a</font>) )
 <b>by </b><i><a class="txt" href="yellow18.html#E2:75_1_5"><i><font color="Green" title="E45">A13</font></i></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/>
<b>thus </b><font color="Maroon" title="c4">g</font> <a href="zf_fund1.html#K1" title="ZF_FUND1:func.1">(#)</a> <font color="Maroon" title="c2">f</font> = 
<font color="Maroon" title="c2">f</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c5">h</font>
<b>by </b><i><a class="txt" href="yellow18.html#E5:75_1_5"><i><font color="Green" title="E46">A14</font></i></a>, <a class="ref" href="yellow16.html#T1" title="YELLOW16:th.1">YELLOW16:1</a></i>
<br/>.= 
<font color="Maroon" title="c4">g</font>
<b>by </b><i><a class="txt" href="yellow18.html#E5:75_1_5"><i><font color="Green" title="E46">A14</font></i></a>, <a class="ref" href="relat_1.html#T79" title="RELAT_1:th.79">RELAT_1:79</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>consider </b><font color="Maroon" title="c1">C</font> being  <a href="altcat_1.html#V6" title="ALTCAT_1:attr.6">strict</a> <a href="altcat_1.html#NM4" title="ALTCAT_1:NM.4">category</a><b> such that </b><br/><a NAME="E7:75_1"/><i><font color="Green" title="E45">A15</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>and </b><br/><a NAME="E8:75_1"/><i><font color="Green" title="E46">A16</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" title="c1">C</font> holds  <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#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>)
 <b>and </b><br/><a NAME="E9:75_1"/><i><font color="Green" title="E47">A17</font></i>: 
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="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">C</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>  &amp; <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="b3">c</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="b1">a</font>,<font color="Olive" title="b2">b</font><br/> for <font color="Olive" title="b5">g</font> being  <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font> holds  <font color="Olive" title="b5">g</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Olive" title="b4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">f</font>,<font color="Olive" title="b5">g</font>)
 <b>from </b><i><a class="ref" href="yellow18.html#S4" target="_self" title="YELLOW18:sch.4">YELLOW18:sch 4</a>(<a class="txt" href="yellow18.html#E2:75_1"><i><font color="Green" title="E41">A6</font></i></a>, <a class="txt" href="yellow18.html#E3:75_1"><i><font color="Green" title="E42">A8</font></i></a>, <a class="txt" href="yellow18.html#E4:75_1"><i><font color="Green" title="E43">A9</font></i></a>, <a class="txt" href="yellow18.html#E5:75_1"><i><font color="Green" title="E44">A12</font></i></a>);<br/></i>
<b>consider </b><font color="Maroon" title="c2">D</font> being   <a href="struct_0.html#NM8" title="STRUCT_0:NM.8">ManySortedSet</a> of <font color="Maroon" title="c1">C</font><b> such that </b><br/><a NAME="E11:75_1"/><i><font color="Green" title="E48">A18</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">C</font> holds  <font color="Maroon" title="c2">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b1">x</font>)
 <b>from </b><i><a class="ref" href="pboole.html#S5" title="PBOOLE:sch.5">PBOOLE:sch 5</a>();<br/></i>
<a NAME="E12:75_1"/><i><font color="Green" title="E49">A19</font></i>: 
<font color="Maroon" title="c1">C</font> is <a href="yellow18.html#V2" title="YELLOW18:attr.2">para-functional</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_6"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon" title="c2">D</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow18.html#D7" target="_self" title="YELLOW18:def.7">YELLOW18:def 7</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">a1</font>, <font color="Olive" title="b2">a2</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">C</font> holds  <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Olive" title="b1">a1</font>,<font color="Olive" title="b2">a2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">a2</font></span>)</span></span><br/>

<b>let </b><font color="Maroon" title="c3">a1</font>, <font color="Maroon" title="c4">a2</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">C</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="c3">a1</font>,<font color="Maroon" title="c4">a2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">a1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a2</font></span>)</span></span><br/>

<a NAME="E1:75_1_6"/>
( <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">a2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">a2</font>) &amp; <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c3">a1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">a1</font> &amp; <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c4">a2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a2</font> )
 
<b>by </b><i><a class="txt" href="yellow18.html#E8:75_1"><i><font color="Green" title="E46">A16</font></i></a>, <a class="txt" href="yellow18.html#E11:75_1"><i><font color="Green" title="E48">A18</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:75_1_6"/>
<span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">a2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">a1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a2</font></span>)</span>
 <b>by </b><i><a class="txt" href="yellow18.html#E2:75"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="yellow18.html#E7:75_1"><i><font color="Green" title="E45">A15</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="E13:75_1"/><i><font color="Green" title="E50">A20</font></i>: 
<font color="Maroon" title="c1">C</font> is <a href="altcat_1.html#V9" title="ALTCAT_1:attr.9">semi-functional</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">a1</font>, <font color="Maroon" title="c4">a2</font>, <font color="Maroon" title="c5">a3</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">C</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="altcat_1.html#D14" title="ALTCAT_1:def.14">ALTCAT_1:def 14</a> <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="c3">a1</font>,<font color="Maroon" title="c4">a2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c5">a3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c5">a3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">a2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span><br/> for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c5">a3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span><br/> for <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>3</sub></font> or  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>4</sub></font> or <font color="Olive">b<sub>2</sub></font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>3</sub></font> <a href="relat_1.html#K5" title="RELAT_1:func.5">*</a> <font color="Olive">b<sub>4</sub></font> ) )</span><br/>

<b>assume </b><a NAME="E1:75_1_7"/><i><font color="Green" title="E51">A21</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="c3">a1</font>,<font color="Maroon" title="c4">a2</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>  &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c5">a3</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>  &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c5">a3</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">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">a2</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span><br/> for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c5">a3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span><br/> for <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>3</sub></font> or  not <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>4</sub></font> or <font color="Olive">b<sub>2</sub></font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>3</sub></font> <a href="relat_1.html#K5" title="RELAT_1:func.5">*</a> <font color="Olive">b<sub>4</sub></font> )</span><br/>

<b>let </b><font color="Maroon" title="c6">f</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c4">a2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c5">a3</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span><br/> for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> or  not <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>3</sub></font> or <font color="Olive">b<sub>1</sub></font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="relat_1.html#K5" title="RELAT_1:func.5">*</a> <font color="Olive">b<sub>3</sub></font> )</span><br/><b>let </b><font color="Maroon" title="c7">g</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c5">a3</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   holds <br/>(  not <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> or  not <font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> or <font color="Maroon" title="c7">g</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="relat_1.html#K5" title="RELAT_1:func.5">*</a> <font color="Olive">b<sub>2</sub></font> )</span><br/><b>let </b><font color="Maroon" title="c8">f'</font>, <font color="Maroon" title="c9">g'</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f'</font> or  not <font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g'</font> or <font color="Maroon" title="c7">g</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f'</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">*</a> <font color="Maroon" title="c9">g'</font> )</span><br/>





<b>assume </b><a NAME="E2:75_1_7"/>
( <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f'</font> &amp; <font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">g'</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">g</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f'</font> <a href="relat_1.html#K5" title="RELAT_1:func.5">*</a> <font color="Maroon" title="c9">g'</font></span><br/>

<b>hence </b><font color="Maroon" title="c7">g</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c6">f</font> = 
<font color="Maroon" title="c8">f'</font> <a href="zf_fund1.html#K1" title="ZF_FUND1:func.1">(#)</a> <font color="Maroon" title="c9">g'</font>
<b>by </b><i><a class="txt" href="yellow18.html#E9:75_1"><i><font color="Green" title="E47">A17</font></i></a>, <a class="txt" href="yellow18.html#E1:75_1_7"><i><font color="Green" title="E51">A21</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c9">g'</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c8">f'</font>
<b>by </b><i><a class="ref" href="yellow16.html#T1" title="YELLOW16:th.1">YELLOW16:1</a></i>
;<br/> <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>
<div><i><font color="Green" title="E51">A22</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">a</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c3">a</font>)</span><br/><a NAME="E1:75_1_8"/>
(  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c3">a</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font>) &amp; <span class="p1"><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">&lt;^</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font>) )
 <b>by </b><i><a class="txt" href="yellow18.html#E3:75"><i><font color="Green" title="E39">A3</font></i></a>, <a class="txt" href="yellow18.html#E7:75_1"><i><font color="Green" title="E45">A15</font></i></a>, <a class="txt" href="yellow18.html#E8:75_1"><i><font color="Green" title="E46">A16</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c4">e</font> =  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c3">a</font>) as   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_8_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">b</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">C</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="c3">a</font>,<font color="Maroon" title="c5">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="c3">a</font>,<font color="Maroon" title="c5">b</font> holds  <font color="Olive" title="b1">f</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> )</span><br/><b>assume </b><a NAME="E1:75_1_8_1"/><i><font color="Green" title="E52">A23</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="c3">a</font>,<font color="Maroon" title="c5">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="c3">a</font>,<font color="Maroon" title="c5">b</font> holds  <font color="Olive" title="b1">f</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font></span><br/><b>let </b><font color="Maroon" title="c6">f</font> be   <a href="altcat_1.html#NM2" title="ALTCAT_1:NM.2">Morphism</a> of <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">b</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">f</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font></span><br/><a NAME="E2:75_1_8_1"/><i><font color="Green" title="E53">A24</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="c3">a</font>,<font color="Maroon" title="c5">b</font></span><a href="altcat_1.html#K1" title="ALTCAT_1:func.1">^&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">b</font>) &amp; <font color="Maroon">F<sub>2</sub></font>(<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">b</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c3">a</font>),<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c5">b</font>) )
 <b>by </b><i><a class="txt" href="yellow18.html#E2:75"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="yellow18.html#E7:75_1"><i><font color="Green" title="E45">A15</font></i></a>, <a class="txt" href="yellow18.html#E8:75_1"><i><font color="Green" title="E46">A16</font></i></a></i>;<br/><a NAME="E3:75_1_8_1"/>
<font color="Maroon" title="c6">f</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="c3">a</font>,<font color="Maroon" title="c5">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="yellow18.html#E1:75_1_8_1"><i><font color="Green" title="E52">A23</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">h</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E5:75_1_8_1"/><i><font color="Green" title="E54">A25</font></i>: 
( <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">h</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c3">a</font>) &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c5">b</font>) )
 <b>by </b><i><a class="txt" href="yellow18.html#E2:75_1_8_1"><i><font color="Green" title="E53">A24</font></i></a>, <a class="ref" href="funct_2.html#D2" title="FUNCT_2:def.2">FUNCT_2:def 2</a></i>;<br/><b>thus </b><font color="Maroon" title="c6">f</font> <a href="altcat_1.html#K5" title="ALTCAT_1:func.5">*</a> <font color="Maroon" title="c4">e</font> = 
<font color="Maroon" title="c7">h</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c3">a</font>)</span>)</span>
<b>by </b><i><a class="txt" href="yellow18.html#E13:75_1"><i><font color="Green" title="E50">A20</font></i></a>, <a class="txt" href="yellow18.html#E1:75_1_8_1"><i><font color="Green" title="E52">A23</font></i></a>, <a class="txt" href="yellow18.html#E5:75_1_8_1"><i><font color="Green" title="E54">A25</font></i></a>, <a class="ref" href="altcat_1.html#D14" title="ALTCAT_1:def.14">ALTCAT_1:def 14</a></i>
<br/>.= 
<font color="Maroon" title="c6">f</font>
<b>by </b><i><a class="txt" href="yellow18.html#E5:75_1_8_1"><i><font color="Green" title="E54">A25</font></i></a>, <a class="ref" href="relat_1.html#T78" title="RELAT_1:th.78">RELAT_1:78</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>hence </b><a NAME="E4:75_1_8"/>
 <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c3">a</font>)
 <b>by </b><i><a class="ref" href="altcat_1.html#D19" title="ALTCAT_1:def.19">ALTCAT_1:def 19</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>
<div><i><font color="Green" title="E52">A26</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_9"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c1">C</font> <a href="yellow18.html#K3" title="YELLOW18:func.3">-carrier_of</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> )</span><br/><b>assume </b><a NAME="E1:75_1_9"/>
<font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">C</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">C</font> <a href="yellow18.html#K3" title="YELLOW18:func.3">-carrier_of</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font></span><br/><b>then reconsider </b><font color="Maroon" title="c4">a</font> = <font color="Maroon" title="c3">i</font> as   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">C</font> ;<br/><b>thus </b><font color="Maroon" title="c1">C</font> <a href="yellow18.html#K3" title="YELLOW18:func.3">-carrier_of</a> <font color="Maroon" title="c3">i</font> = 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <span class="p1">(<span class="default"><a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c4">a</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow18.html#D8" target="_self" title="YELLOW18:def.8">Def8</a></i>
<br/>.= 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c4">a</font>)</span>)</span>
<b>by </b><i><a class="txt" href="yellow18.html#E14:75_1"><i><font color="Green" title="E51">A22</font></i></a></i>
<br/>.= 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c4">a</font>)</span>)</span>

<br/>.= 
<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c4">a</font>)
<b>by </b><i><a class="ref" href="relat_1.html#T71" title="RELAT_1:th.71">RELAT_1:71</a></i>
<br/>.= 
<font color="Maroon" title="c2">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font>
<b>by </b><i><a class="txt" href="yellow18.html#E11:75_1"><i><font color="Green" title="E48">A18</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="E16:75_1"/>
<font color="Maroon" title="c1">C</font> is <a href="yellow18.html#V3" title="YELLOW18:attr.3">set-id-inheriting</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">a</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c1">C</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow18.html#D10" target="_self" title="YELLOW18:def.10">YELLOW18:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><a href="yellow18.html#K3" title="YELLOW18:func.3">the_carrier_of</a> <font color="Maroon" title="c3">a</font></span>)</span></span><br/>

<b>thus </b> <a href="altcat_1.html#K8" title="ALTCAT_1:func.8">idm</a> <font color="Maroon" title="c3">a</font> = 
 <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c3">a</font>)
<b>by </b><i><a class="txt" href="yellow18.html#E14:75_1"><i><font color="Green" title="E51">A22</font></i></a></i>
<br/>.= 
 <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">a</font></span>)</span>
<b>by </b><i><a class="txt" href="yellow18.html#E11:75_1"><i><font color="Green" title="E48">A18</font></i></a></i>
<br/>.= 
 <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><a href="yellow18.html#K3" title="YELLOW18:func.3">the_carrier_of</a> <font color="Maroon" title="c3">a</font></span>)</span>
<b>by </b><i><a class="txt" href="yellow18.html#E15:75_1"><i><font color="Green" title="E52">A26</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>then reconsider </b><font color="Maroon" title="c3">C</font> = <font color="Maroon" title="c1">C</font> as  <a href="altcat_1.html#V6" title="ALTCAT_1:attr.6">strict</a> <a href="altcat_1.html#V9" title="ALTCAT_1:attr.9">semi-functional</a> <a href="yellow18.html#V2" title="YELLOW18:attr.2">para-functional</a> <a href="yellow18.html#V3" title="YELLOW18:attr.3">set-id-inheriting</a> <a href="altcat_1.html#NM4" title="ALTCAT_1:NM.4">category</a> <b>by </b><i><a class="txt" href="yellow18.html#E12:75_1"><i><font color="Green" title="E49">A19</font></i></a>, <a class="txt" href="yellow18.html#E13:75_1"><i><font color="Green" title="E50">A20</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c3">C</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; ( for <font color="Olive" title="b1">a</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c3">C</font> holds   <a href="yellow18.html#K3" title="YELLOW18:func.3">the_carrier_of</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b1">a</font>) ) &amp; ( 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" title="c3">C</font> holds  <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#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>) ) )</span><br/>

<b>thus </b><a NAME="E18:75_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>by </b><i><a class="txt" href="yellow18.html#E7:75_1"><i><font color="Green" title="E45">A15</font></i></a></i>; <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">a</font> being  <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c3">C</font> holds   <a href="yellow18.html#K3" title="YELLOW18:func.3">the_carrier_of</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b1">a</font>) ) &amp; ( 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" title="c3">C</font> holds  <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#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>) ) )</span><br/>

<div><b>hereby </b> <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">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" title="c3">C</font> holds  <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#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>)</span>
<div class="add"><b>let </b><font color="Maroon" title="c4">a</font> be   <a href="altcat_1.html#NM1" title="ALTCAT_1:NM.1">object</a> of <font color="Maroon" title="c3">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="yellow18.html#K3" title="YELLOW18:func.3">the_carrier_of</a> <font color="Maroon" title="c4">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c4">a</font>)</span><br/><b>thus </b> <a href="yellow18.html#K3" title="YELLOW18:func.3">the_carrier_of</a> <font color="Maroon" title="c4">a</font> = 
<font color="Maroon" title="c2">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">a</font>
<b>by </b><i><a class="txt" href="yellow18.html#E15:75_1"><i><font color="Green" title="E52">A26</font></i></a></i>
<br/>.= 
<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c4">a</font>)
<b>by </b><i><a class="txt" href="yellow18.html#E11:75_1"><i><font color="Green" title="E48">A18</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="E20:75_1"/>
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" title="c3">C</font> holds  <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#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>)
 <b>by </b><i><a class="txt" href="yellow18.html#E8:75_1"><i><font color="Green" title="E46">A16</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>
