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<b>let </b><font color="Maroon" title="c1">A</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">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="c3">a</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">D</font>;<br/><b>let </b><font color="Maroon" title="c4">o</font> be   <a href="binop_1.html#NM2" title="BINOP_1:NM.2">BinOp</a> of <font color="Maroon" title="c2">D</font>;<br/><b>let </b><font color="Maroon" title="c5">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">A</font>,<font color="Maroon" title="c2">D</font>;<br/>









<b>assume </b><a NAME="E1:16"/><i><font color="Green" title="E6">A1</font></i>: 
<font color="Maroon" title="c3">a</font> <a href="binop_1.html#R3" title="BINOP_1:pred.3">is_a_unity_wrt</a> <font color="Maroon" title="c4">o</font>
 ;<br/>

<a NAME="E2:16"/><i><font color="Green" title="E7">A2</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="monoid_1.html#K6" title="MONOID_1:func.6">[;]</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="monoid_1.html#K7" title="MONOID_1:func.7">[:]</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c3">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">D</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:16_1"/><i><font color="Green" title="E8">A3</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 ;<br/><a NAME="E2:16_1"/><b>then </b>
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">f</font>
 <b>by </b><i><a class="txt" href="monoid_1.html#E2:16"><i><font color="Green" title="E7">A2</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c7">b</font> = <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">D</font> ;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="monoid_1.html#K6" title="MONOID_1:func.6">[;]</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> = 
<font color="Maroon" title="c4">o</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c7">b</font>
<b>by </b><i><a class="txt" href="monoid_1.html#E2:16"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="monoid_1.html#E1:16_1"><i><font color="Green" title="E8">A3</font></i></a>, <a class="ref" href="funcop_1.html#T42" title="FUNCOP_1:th.42">FUNCOP_1:42</a></i>
<br/>.= 
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font>
<b>by </b><i><a class="txt" href="monoid_1.html#E1:16"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="binop_1.html#T11" title="BINOP_1:th.11">BINOP_1:11</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:16"/>
<font color="Maroon" title="c4">o</font> <a href="monoid_1.html#K6" title="MONOID_1:func.6">[;]</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">f</font>
 <b>by </b><i><a class="txt" href="monoid_1.html#E2:16"><i><font color="Green" title="E7">A2</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:16_2"/><i><font color="Green" title="E8">A4</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 ;<br/><a NAME="E2:16_2"/><b>then </b>
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c5">f</font>
 <b>by </b><i><a class="txt" href="monoid_1.html#E2:16"><i><font color="Green" title="E7">A2</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c7">b</font> = <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">D</font> ;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c4">o</font> <a href="monoid_1.html#K7" title="MONOID_1:func.7">[:]</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c3">a</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font> = 
<font color="Maroon" title="c4">o</font> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c3">a</font>
<b>by </b><i><a class="txt" href="monoid_1.html#E2:16"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="monoid_1.html#E1:16_2"><i><font color="Green" title="E8">A4</font></i></a>, <a class="ref" href="funcop_1.html#T35" title="FUNCOP_1:th.35">FUNCOP_1:35</a></i>
<br/>.= 
<font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">x</font>
<b>by </b><i><a class="txt" href="monoid_1.html#E1:16"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="binop_1.html#T11" title="BINOP_1:th.11">BINOP_1:11</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:16"/>
<font color="Maroon" title="c4">o</font> <a href="monoid_1.html#K7" title="MONOID_1:func.7">[:]</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c3">a</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c5">f</font>
 <b>by </b><i><a class="txt" href="monoid_1.html#E2:16"><i><font color="Green" title="E7">A2</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>


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