<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">I</font>, <font color="Maroon" title="c2">J</font>, <font color="Maroon" title="c3">D</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c4">F</font>, <font color="Maroon" title="c5">G</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c3">D</font>;<br/><b>let </b><font color="Maroon" title="c6">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c3">D</font>;<br/><b>let </b><font color="Maroon" title="c7">g</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c3">D</font>;<br/><b>let </b><font color="Maroon" title="c8">X</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c1">I</font>;<br/><b>let </b><font color="Maroon" title="c9">Y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="finsub_1.html#K5" title="FINSUB_1:func.5">Fin</a> <font color="Maroon" title="c2">J</font>;<br/>











<b>assume </b><a NAME="E1:33"/><i><font color="Green" title="E14">A1</font></i>: 
( <font color="Maroon" title="c4">F</font> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a> &amp; <font color="Maroon" title="c4">F</font> is <a href="binop_1.html#V2" title="BINOP_1:attr.2">associative</a> &amp; ( <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c9">Y</font>,<font color="Maroon" title="c8">X</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</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>  or <font color="Maroon" title="c4">F</font> <a href="setwiseo.html#NR1" title="SETWISEO:NR.1">has_a_unity</a>  ) )
 ;<br/>

<b>assume </b><a NAME="E2:33"/><i><font color="Green" title="E15">A2</font></i>: 
<font color="Maroon" title="c5">G</font> is <a href="binop_1.html#V1" title="BINOP_1:attr.1">commutative</a>
 ;<br/>

<a NAME="E3:33"/><i><font color="Green" title="E16">A3</font></i>: 
( <font color="Maroon" title="c8">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">I</font> &amp; <font color="Maroon" title="c9">Y</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">J</font> )
 
<b>by </b><i><a class="ref" href="finsub_1.html#D5" title="FINSUB_1:def.5">FINSUB_1:def 5</a></i>;<br/>
<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> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">$2,$1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>;<br/>
<b>consider </b><font color="Maroon" title="c10">h</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E5:33"/><i><font color="Green" title="E17">A4</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c10">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c1">I</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; ( for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">J</font> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> holds <br/><font color="Maroon" title="c10">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>) ) )
 <b>from </b><i><a class="ref" href="funct_3.html#S2" title="FUNCT_3:sch.2">FUNCT_3:sch 2</a>();<br/></i>
<a NAME="E6:33"/><i><font color="Green" title="E18">A5</font></i>: 
<font color="Maroon" title="c10">h</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="33_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c11">z1</font>, <font color="Maroon" title="c12">z2</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:33_1_1"/><i><font color="Green" title="E19">A6</font></i>: 
( <font color="Maroon" title="c11">z1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c10">h</font> &amp; <font color="Maroon" title="c12">z2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c10">h</font> &amp; <font color="Maroon" title="c10">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">z2</font> )
 ;<br/><b>then consider </b><font color="Maroon" title="c13">x1</font>, <font color="Maroon" title="c14">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:33_1_1"/><i><font color="Green" title="E20">A7</font></i>: 
<font color="Maroon" title="c11">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">x1</font>,<font color="Maroon" title="c14">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="matrix_3.html#E5:33"><i><font color="Green" title="E17">A4</font></i></a>, <a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></i>;<br/><b>consider </b><font color="Maroon" title="c15">x2</font>, <font color="Maroon" title="c16">y2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:33_1_1"/><i><font color="Green" title="E21">A8</font></i>: 
<font color="Maroon" title="c12">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">x2</font>,<font color="Maroon" title="c16">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="matrix_3.html#E5:33"><i><font color="Green" title="E17">A4</font></i></a>, <a class="txt" href="matrix_3.html#E1:33_1_1"><i><font color="Green" title="E19">A6</font></i></a>, <a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></i>;<br/><a NAME="E6:33_1_1"/><i><font color="Green" title="E22">A9</font></i>: 
( <font color="Maroon" title="c13">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">J</font> &amp; <font color="Maroon" title="c14">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> &amp; <font color="Maroon" title="c15">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">J</font> &amp; <font color="Maroon" title="c16">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> )
 <b>by </b><i><a class="txt" href="matrix_3.html#E5:33"><i><font color="Green" title="E17">A4</font></i></a>, <a class="txt" href="matrix_3.html#E1:33_1_1"><i><font color="Green" title="E19">A6</font></i></a>, <a class="txt" href="matrix_3.html#E3:33_1_1"><i><font color="Green" title="E20">A7</font></i></a>, <a class="txt" href="matrix_3.html#E5:33_1_1"><i><font color="Green" title="E21">A8</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E7:33_1_1"/><b>then </b><i><font color="Green" title="E23">A10</font></i>: 
<font color="Maroon" title="c10">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c13">x1</font>,<font color="Maroon" title="c14">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">y1</font>,<font color="Maroon" title="c13">x1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="matrix_3.html#E5:33"><i><font color="Green" title="E17">A4</font></i></a></i>;<br/><a NAME="E8:33_1_1"/>
<font color="Maroon" title="c10">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c15">x2</font>,<font color="Maroon" title="c16">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">y2</font>,<font color="Maroon" title="c15">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="matrix_3.html#E5:33"><i><font color="Green" title="E17">A4</font></i></a>, <a class="txt" href="matrix_3.html#E6:33_1_1"><i><font color="Green" title="E22">A9</font></i></a></i>;<br/><a NAME="E9:33_1_1"/><b>then </b>
( <font color="Maroon" title="c14">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">y2</font> &amp; <font color="Maroon" title="c13">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">x2</font> )
 <b>by </b><i><a class="txt" href="matrix_3.html#E3:33_1_1"><i><font color="Green" title="E20">A7</font></i></a>, <a class="txt" href="matrix_3.html#E5:33_1_1"><i><font color="Green" title="E21">A8</font></i></a>, <a class="txt" href="matrix_3.html#E1:33_1_1"><i><font color="Green" title="E19">A6</font></i></a>, <a class="txt" href="matrix_3.html#E7:33_1_1"><i><font color="Green" title="E23">A10</font></i></a>, <a class="ref" href="zfmisc_1.html#T33" title="ZFMISC_1:th.33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E10:33_1_1"/>
<font color="Maroon" title="c11">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">z2</font>
 <b>by </b><i><a class="txt" href="matrix_3.html#E3:33_1_1"><i><font color="Green" title="E20">A7</font></i></a>, <a class="txt" href="matrix_3.html#E5:33_1_1"><i><font color="Green" title="E21">A8</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:33_1"/>
<font color="Maroon" title="c10">h</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:33"/><i><font color="Green" title="E19">A11</font></i>: 
<font color="Maroon" title="c10">h</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c9">Y</font>,<font color="Maroon" title="c8">X</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c8">X</font>,<font color="Maroon" title="c9">Y</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2"><b>proof </b></a><div class="add">

<a NAME="E1:33_2"/>
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c8">X</font>,<font color="Maroon" title="c9">Y</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span> iff <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">h</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c9">Y</font>,<font color="Maroon" title="c8">X</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>thus </b><a NAME="E1:33_2_1"/>
( <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c8">X</font>,<font color="Maroon" title="c9">Y</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span> implies <font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">h</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c9">Y</font>,<font color="Maroon" title="c8">X</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:33_2_1_1"/><i><font color="Green" title="E20">A12</font></i>: 
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c8">X</font>,<font color="Maroon" title="c9">Y</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">y1</font>, <font color="Maroon" title="c13">x1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:33_2_1_1"/><i><font color="Green" title="E21">A13</font></i>: 
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">y1</font>,<font color="Maroon" title="c13">x1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></i>;<br/>
<a NAME="E4:33_2_1_1"/><i><font color="Green" title="E22">A14</font></i>: 
( <font color="Maroon" title="c12">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font> &amp; <font color="Maroon" title="c13">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Y</font> )
 
<b>by </b><i><a class="txt" href="matrix_3.html#E1:33_2_1_1"><i><font color="Green" title="E20">A12</font></i></a>, <a class="txt" href="matrix_3.html#E3:33_2_1_1"><i><font color="Green" title="E21">A13</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E5:33_2_1_1"/><b>then </b><i><font color="Green" title="E23">A15</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">x1</font>,<font color="Maroon" title="c12">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c9">Y</font>,<font color="Maroon" title="c8">X</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E6:33_2_1_1"/><i><font color="Green" title="E24">A16</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">x1</font>,<font color="Maroon" title="c12">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c10">h</font>
 
<b>by </b><i><a class="txt" href="matrix_3.html#E3:33"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="matrix_3.html#E5:33"><i><font color="Green" title="E17">A4</font></i></a>, <a class="txt" href="matrix_3.html#E4:33_2_1_1"><i><font color="Green" title="E22">A14</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E7:33_2_1_1"/>
<font color="Maroon" title="c10">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c13">x1</font>,<font color="Maroon" title="c12">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y</font>
 
<b>by </b><i><a class="txt" href="matrix_3.html#E3:33"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="matrix_3.html#E5:33"><i><font color="Green" title="E17">A4</font></i></a>, <a class="txt" href="matrix_3.html#E3:33_2_1_1"><i><font color="Green" title="E21">A13</font></i></a>, <a class="txt" href="matrix_3.html#E4:33_2_1_1"><i><font color="Green" title="E22">A14</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:33_2_1_1"/>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">h</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c9">Y</font>,<font color="Maroon" title="c8">X</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>
 <b>by </b><i><a class="txt" href="matrix_3.html#E5:33_2_1_1"><i><font color="Green" title="E23">A15</font></i></a>, <a class="txt" href="matrix_3.html#E6:33_2_1_1"><i><font color="Green" title="E24">A16</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:33_2_1"/>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">h</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c9">Y</font>,<font color="Maroon" title="c8">X</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:33_2_1"/><i><font color="Green" title="E20">A17</font></i>: 
( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c10">h</font> &amp; <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c9">Y</font>,<font color="Maroon" title="c8">X</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span> &amp; <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c13">x1</font>, <font color="Maroon" title="c14">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:33_2_1"/><i><font color="Green" title="E21">A18</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">x1</font>,<font color="Maroon" title="c14">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="txt" href="matrix_3.html#E4:33_2_1"><i><font color="Green" title="E20">A17</font></i></a>, <a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></i>;<br/>
<a NAME="E7:33_2_1"/><i><font color="Green" title="E22">A19</font></i>: 
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c13">x1</font>,<font color="Maroon" title="c14">y1</font>
 
<b>by </b><i><a class="txt" href="matrix_3.html#E4:33_2_1"><i><font color="Green" title="E20">A17</font></i></a>, <a class="txt" href="matrix_3.html#E6:33_2_1"><i><font color="Green" title="E21">A18</font></i></a></i>;<br/>
<a NAME="E8:33_2_1"/><i><font color="Green" title="E23">A20</font></i>: 
( <font color="Maroon" title="c13">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">Y</font> &amp; <font color="Maroon" title="c14">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">X</font> )
 
<b>by </b><i><a class="txt" href="matrix_3.html#E4:33_2_1"><i><font color="Green" title="E20">A17</font></i></a>, <a class="txt" href="matrix_3.html#E6:33_2_1"><i><font color="Green" title="E21">A18</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E9:33_2_1"/><b>then </b>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">y1</font>,<font color="Maroon" title="c13">x1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="matrix_3.html#E3:33"><i><font color="Green" title="E16">A3</font></i></a>, <a class="txt" href="matrix_3.html#E5:33"><i><font color="Green" title="E17">A4</font></i></a>, <a class="txt" href="matrix_3.html#E7:33_2_1"><i><font color="Green" title="E22">A19</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:33_2_1"/>
<font color="Maroon" title="c11">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c8">X</font>,<font color="Maroon" title="c9">Y</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>
 <b>by </b><i><a class="txt" href="matrix_3.html#E8:33_2_1"><i><font color="Green" title="E23">A20</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:33_2"/>
<font color="Maroon" title="c10">h</font> <a href="relat_1.html#K9" title="RELAT_1:func.9">.:</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c9">Y</font>,<font color="Maroon" title="c8">X</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c8">X</font>,<font color="Maroon" title="c9">Y</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:33"/><i><font color="Green" title="E20">A21</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c1">I</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> holds <br/><font color="Maroon" title="c10">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c2">J</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:33_3"/><i><font color="Green" title="E21">A22</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c1">I</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">y</font>, <font color="Maroon" title="c13">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:33_3"/><i><font color="Green" title="E22">A23</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c13">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></i>;<br/>
<a NAME="E4:33_3"/><i><font color="Green" title="E23">A24</font></i>: 
( <font color="Maroon" title="c12">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">J</font> &amp; <font color="Maroon" title="c13">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> )
 
<b>by </b><i><a class="txt" href="matrix_3.html#E1:33_3"><i><font color="Green" title="E21">A22</font></i></a>, <a class="txt" href="matrix_3.html#E3:33_3"><i><font color="Green" title="E22">A23</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E5:33_3"/><b>then </b>
<font color="Maroon" title="c10">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c12">y</font>,<font color="Maroon" title="c13">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">z</font>,<font color="Maroon" title="c12">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="matrix_3.html#E5:33"><i><font color="Green" title="E17">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:33_3"/>
<font color="Maroon" title="c10">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c2">J</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="matrix_3.html#E3:33_3"><i><font color="Green" title="E22">A23</font></i></a>, <a class="txt" href="matrix_3.html#E4:33_3"><i><font color="Green" title="E23">A24</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:33"/><i><font color="Green" title="E21">A25</font></i>: 
<font color="Maroon" title="c5">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c7">g</font>,<font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c10">h</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_4"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c11">G</font> = <font color="Maroon" title="c5">G</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">D</font>,<font color="Maroon" title="c3">D</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<font color="Maroon" title="c3">D</font> ;<br/>
<a NAME="E2:33_4"/><i><font color="Green" title="E22">A26</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c2">J</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E3:33_4"/><i><font color="Green" title="E23">A27</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c7">g</font>,<font color="Maroon" title="c6">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c1">I</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:33_4"/>
( ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c7">g</font>,<font color="Maroon" title="c6">f</font></span>)</span> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c10">h</font> &amp; <font color="Maroon" title="c10">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font></span>)</span> ) ) ) &amp; ( for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c7">g</font>,<font color="Maroon" title="c6">f</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c11">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c7">g</font>,<font color="Maroon" title="c6">f</font></span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_4_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:33_4_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c7">g</font>,<font color="Maroon" title="c6">f</font></span>)</span> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c10">h</font> &amp; <font color="Maroon" title="c10">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font></span>)</span> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_4_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>thus </b><a NAME="E1:33_4_1_1"/>
( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c7">g</font>,<font color="Maroon" title="c6">f</font></span>)</span> implies ( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c10">h</font> &amp; <font color="Maroon" title="c10">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font></span>)</span> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_4_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:33_4_1_1_1"/><i><font color="Green" title="E24">A28</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c7">g</font>,<font color="Maroon" title="c6">f</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c13">y</font>, <font color="Maroon" title="c14">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:33_4_1_1_1"/><i><font color="Green" title="E25">A29</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c14">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></i>;<br/>
<a NAME="E4:33_4_1_1_1"/><i><font color="Green" title="E26">A30</font></i>: 
( <font color="Maroon" title="c13">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">J</font> &amp; <font color="Maroon" title="c14">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">I</font> )
 
<b>by </b><i><a class="txt" href="matrix_3.html#E3:33_4"><i><font color="Green" title="E23">A27</font></i></a>, <a class="txt" href="matrix_3.html#E1:33_4_1_1_1"><i><font color="Green" title="E24">A28</font></i></a>, <a class="txt" href="matrix_3.html#E3:33_4_1_1_1"><i><font color="Green" title="E25">A29</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E5:33_4_1_1_1"/><b>then </b>
<font color="Maroon" title="c10">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c14">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">z</font>,<font color="Maroon" title="c13">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<b>by </b><i><a class="txt" href="matrix_3.html#E5:33"><i><font color="Green" title="E17">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:33_4_1_1_1"/>
( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c10">h</font> &amp; <font color="Maroon" title="c10">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font></span>)</span> )
 <b>by </b><i><a class="txt" href="matrix_3.html#E5:33"><i><font color="Green" title="E17">A4</font></i></a>, <a class="txt" href="matrix_3.html#E2:33_4"><i><font color="Green" title="E22">A26</font></i></a>, <a class="txt" href="matrix_3.html#E4:33_4_1_1_1"><i><font color="Green" title="E26">A30</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a>, <a class="txt" href="matrix_3.html#E3:33_4_1_1_1"><i><font color="Green" title="E25">A29</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:33_4_1_1"/>
( <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c10">h</font> &amp; <font color="Maroon" title="c10">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font></span>)</span> )
 ;<br/>

<b>hence </b><a NAME="E3:33_4_1_1"/>
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c7">g</font>,<font color="Maroon" title="c6">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="matrix_3.html#E5:33"><i><font color="Green" title="E17">A4</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:33_4_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c7">g</font>,<font color="Maroon" title="c6">f</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c11">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c7">g</font>,<font color="Maroon" title="c6">f</font></span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_4_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c12">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:33_4_1_2"/><i><font color="Green" title="E24">A31</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c7">g</font>,<font color="Maroon" title="c6">f</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c13">y</font>, <font color="Maroon" title="c14">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:33_4_1_2"/><i><font color="Green" title="E25">A32</font></i>: 
<font color="Maroon" title="c12">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c13">y</font>,<font color="Maroon" title="c14">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <b>by </b><i><a class="ref" href="relat_1.html#D1" title="RELAT_1:def.1">RELAT_1:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">y</font> = <font color="Maroon" title="c13">y</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">J</font> <b>by </b><i><a class="txt" href="matrix_3.html#E3:33_4"><i><font color="Green" title="E23">A27</font></i></a>, <a class="txt" href="matrix_3.html#E1:33_4_1_2"><i><font color="Green" title="E24">A31</font></i></a>, <a class="txt" href="matrix_3.html#E3:33_4_1_2"><i><font color="Green" title="E25">A32</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">z</font> = <font color="Maroon" title="c14">z</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">I</font> <b>by </b><i><a class="txt" href="matrix_3.html#E3:33_4"><i><font color="Green" title="E23">A27</font></i></a>, <a class="txt" href="matrix_3.html#E1:33_4_1_2"><i><font color="Green" title="E24">A31</font></i></a>, <a class="txt" href="matrix_3.html#E3:33_4_1_2"><i><font color="Green" title="E25">A32</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E6:33_4_1_2"/><i><font color="Green" title="E26">A33</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">z</font>,<font color="Maroon" title="c15">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">y</font>,<font color="Maroon" title="c16">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c7">g</font>,<font color="Maroon" title="c6">f</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="matrix_3.html#E2:33_4"><i><font color="Green" title="E22">A26</font></i></a>, <a class="txt" href="matrix_3.html#E3:33_4"><i><font color="Green" title="E23">A27</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c11">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">h</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c15">y</font>,<font color="Maroon" title="c16">z</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix_3.html#E3:33_4_1_2"><i><font color="Green" title="E25">A32</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c16">z</font>,<font color="Maroon" title="c15">y</font>
<b>by </b><i><a class="txt" href="matrix_3.html#E5:33"><i><font color="Green" title="E17">A4</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c11">G</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">z</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">y</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix_3.html#E6:33_4_1_2"><i><font color="Green" title="E26">A33</font></i></a>, <a class="ref" href="finseqop.html#T82" title="FINSEQOP:th.82">FINSEQOP:82</a></i>
<br/>.= 
<font color="Maroon" title="c11">G</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">y</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">z</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix_3.html#E2:33"><i><font color="Green" title="E15">A2</font></i></a>, <a class="ref" href="binop_1.html#D2" title="BINOP_1:def.2">BINOP_1:def 2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c7">g</font>,<font color="Maroon" title="c6">f</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c15">y</font>,<font color="Maroon" title="c16">z</font>
<b>by </b><i><a class="txt" href="matrix_3.html#E6:33_4_1_2"><i><font color="Green" title="E26">A33</font></i></a>, <a class="ref" href="finseqop.html#T82" title="FINSEQOP:th.82">FINSEQOP:82</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c7">g</font>,<font color="Maroon" title="c6">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">x</font>
<b>by </b><i><a class="txt" href="matrix_3.html#E3:33_4_1_2"><i><font color="Green" title="E25">A32</font></i></a></i>
;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:33_4_1"/>
verum
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:33_4"/>
<font color="Maroon" title="c5">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c7">g</font>,<font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c10">h</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T20" title="FUNCT_1:th.20">FUNCT_1:20</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c11">h</font> = <font color="Maroon" title="c10">h</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c2">J</font>,<font color="Maroon" title="c1">I</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c1">I</font>,<font color="Maroon" title="c2">J</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <b>by </b><i><a class="txt" href="matrix_3.html#E5:33"><i><font color="Green" title="E17">A4</font></i></a>, <a class="txt" href="matrix_3.html#E8:33"><i><font color="Green" title="E20">A21</font></i></a>, <a class="ref" href="funct_2.html#T5" title="FUNCT_2:th.5">FUNCT_2:5</a></i>;<br/>
<font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c8">X</font>,<font color="Maroon" title="c9">Y</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font></span>)</span> = 
<font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c9">Y</font>,<font color="Maroon" title="c8">X</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c11">h</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix_3.html#E1:33"><i><font color="Green" title="E14">A1</font></i></a>, <a class="txt" href="matrix_3.html#E6:33"><i><font color="Green" title="E18">A5</font></i></a>, <a class="txt" href="matrix_3.html#E7:33"><i><font color="Green" title="E19">A11</font></i></a>, <a class="ref" href="setwop_2.html#T8" title="SETWOP_2:th.8">SETWOP_2:8</a></i>
<br/>.= 
<font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c9">Y</font>,<font color="Maroon" title="c8">X</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c7">g</font>,<font color="Maroon" title="c6">f</font></span>)</span>
<b>by </b><i><a class="txt" href="matrix_3.html#E9:33"><i><font color="Green" title="E21">A25</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E12:33"/>
<font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c8">X</font>,<font color="Maroon" title="c9">Y</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c6">f</font>,<font color="Maroon" title="c7">g</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="setwiseo.html#K7" title="SETWISEO:func.7">$$</a> <span class="p1"><a href="matrix_3.html#K8" title="MATRIX_3:func.8">[:</a><span class="default"><font color="Maroon" title="c9">Y</font>,<font color="Maroon" title="c8">X</font></span><a href="matrix_3.html#K8" title="MATRIX_3:func.8">:]</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_3.html#K9" title="MATRIX_3:func.9">*</a> <font color="Maroon" title="c7">g</font>,<font color="Maroon" title="c6">f</font></span>)</span>
 ;<br/>


</div>
