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<b>set </b><font color="Maroon">c<sub>1</sub></font> = <a href="binop_2.html#K35">multreal</a>  <a href="realset1.html#K1">||</a> <a href="numbers.html#K5">NAT</a> ;<br/>
<a NAME="E1:119_1_1"/>
( <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><a href="numbers.html#K5">NAT</a> ,<a href="numbers.html#K5">NAT</a> </span><a href="mcart_1.html#K12">:]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<a href="numbers.html#K1">REAL</a> </span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <a href="relat_1.html#K1">dom</a> <a href="binop_2.html#K35">multreal</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<a href="numbers.html#K1">REAL</a> </span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E2:119_1_1"/><b>then </b><i><font color="Green">E55</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="binop_2.html#K35">multreal</a>  <a href="realset1.html#K1">||</a> <a href="numbers.html#K5">NAT</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K12">[:</a><span class="default"><a href="numbers.html#K5">NAT</a> ,<a href="numbers.html#K5">NAT</a> </span><a href="mcart_1.html#K12">:]</a></span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T91">RELAT_1:91</a></i>;<br/>
<a NAME="E3:119_1_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="binop_2.html#K35">multreal</a>  <a href="realset1.html#K1">||</a> <a href="numbers.html#K5">NAT</a> </span>)</span> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="119_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:119_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="binop_2.html#K35">multreal</a>  <a href="realset1.html#K1">||</a> <a href="numbers.html#K5">NAT</a> </span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:119_1_1_1"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="binop_2.html#K35">multreal</a>  <a href="realset1.html#K1">||</a> <a href="numbers.html#K5">NAT</a> </span>)</span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="binop_2.html#K35">multreal</a>  <a href="realset1.html#K1">||</a> <a href="numbers.html#K5">NAT</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="mcart_1.html#K1">`1</a> , <font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="mcart_1.html#K2">`2</a>  as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="monoid_0.html#E2:119_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="monoid_0.html#E3:119_1_1_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/>
<font color="Maroon">c<sub>2</sub></font> = 
<a href="binop_2.html#K35">multreal</a>  <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="monoid_0.html#E3:119_1_1_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>
<br/>.= 
<a href="binop_2.html#K35">multreal</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="domain_1.html#K1">]</a></span>
<b>by </b><i><a class="txt" href="monoid_0.html#E2:119_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="monoid_0.html#E3:119_1_1_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="mcart_1.html#T23">MCART_1:23</a></i>
<br/>.= 
<a href="binop_2.html#K35">multreal</a>  <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>

<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="binop_2.html#K24">*</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="binop_2.html#D11">BINOP_2:def 11</a></i>
;<br/>
<b>hence </b><a NAME="E6:119_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>2</sub></font> = <a href="binop_2.html#K35">multreal</a>  <a href="realset1.html#K1">||</a> <a href="numbers.html#K5">NAT</a>  as   <a href="binop_1.html#NM2">BinOp</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="monoid_0.html#E2:119_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<a NAME="E5:119_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_4.html#NR1" target="_self">&lt;=</a> <font color="Maroon">H<sub>2</sub></font>( <a href="monoid_0.html#K7" target="_self">&lt;REAL,*&gt;</a> )
 
<b>by </b><i><a class="ref" href="relat_1.html#T88">RELAT_1:88</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> =  <a href="group_1.html#G1">HGrStr</a>(# <a href="numbers.html#K5">NAT</a> ,<font color="Maroon">c<sub>2</sub></font> #) as  non <a href="struct_0.html#V3">empty</a> <a href="group_1.html#V1">strict</a>  <a href="monoid_0.html#M2" target="_self">SubStr</a> of  <a href="monoid_0.html#K7" target="_self">&lt;REAL,*&gt;</a>  <b>by </b><i><a class="ref" href="monoid_0.html#D23" target="_self">Def23</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = 1 as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font> ;<br/>
<div><i><font color="Green">E56</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="119_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/><b>thus </b><font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>5</sub></font> = 
1 <a href="binop_2.html#K24">*</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="ref" href="monoid_0.html#T58" target="_self">Th58</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font>

;<br/><b>thus </b><font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>4</sub></font> = 
<font color="Maroon">c<sub>6</sub></font> <a href="binop_2.html#K24">*</a> 1
<b>by </b><i><a class="ref" href="monoid_0.html#T58" target="_self">Th58</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font>

;<br/></div><b>end;</b></div>
<a NAME="E9:119_1_1"/><b>then </b><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="group_1.html#V2">unital</a>
 
<b>by </b><i><a class="ref" href="monoid_0.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E10:119_1_1"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="setwiseo.html#NR1" target="_self">has_a_unity</a> 
 
<b>by </b><i><a class="ref" href="monoid_0.html#D10" target="_self">Def10</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="119_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font>;<br/><b>thus </b><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> = 
<font color="Maroon">c<sub>4</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>5</sub></font>

<br/>.= 
<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="monoid_0.html#E8:119_1_1"><i><font color="Green">E56</font></i></a></i>
;<br/><b>thus </b><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font> = 
<font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>4</sub></font>

<br/>.= 
<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="monoid_0.html#E8:119_1_1"><i><font color="Green">E56</font></i></a></i>
;<br/></div><b>end;</b></div>
<a NAME="E12:119_1_1"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="binop_1.html#R3">is_a_unity_wrt</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>3</sub></font>)
 
<b>by </b><i><a class="ref" href="binop_1.html#T11">BINOP_1:11</a></i>;<br/>
<a NAME="E13:119_1_1"/><b>then </b><i><font color="Green">E60</font></i>: 
 <a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>3</sub></font>) <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="binop_1.html#D8">BINOP_1:def 8</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="119_1_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>3</sub></font>;<br/><b>assume </b><a NAME="E1:119_1_1_4"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>3</sub></font>)
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E3:119_1_1_4"/>
<font color="Maroon">c<sub>5</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="binop_2.html#K24">*</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="monoid_0.html#T58" target="_self">Th58</a></i>;<br/><a NAME="E4:119_1_1_4"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="txt" href="monoid_0.html#E13:119_1_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="monoid_0.html#E1:119_1_1_4"><i><font color="Green">E61</font></i></a>, <a class="ref" href="nat_1.html#T40">NAT_1:40</a></i>;<br/><b>hence </b><a NAME="E5:119_1_1_4"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="binop_1.html#K3">the_unity_wrt</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>3</sub></font>) )
 <b>by </b><i><a class="txt" href="monoid_0.html#E12:119_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="binop_1.html#D8">BINOP_1:def 8</a></i>;<br/></div><b>end;</b></div>
<a NAME="E15:119_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> is <a href="monoid_0.html#V17" target="_self">uniquely-decomposable</a>
 
<b>by </b><i><a class="txt" href="monoid_0.html#E10:119_1_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="monoid_0.html#T16" target="_self">Th16</a></i>;<br/>
<b>hence </b><a NAME="E16:119_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being non <a href="struct_0.html#V3">empty</a> <a href="group_1.html#V1">strict</a> <a href="group_1.html#V2">unital</a> <a href="monoid_0.html#V17" target="_self">uniquely-decomposable</a>  <a href="monoid_0.html#M2" target="_self">SubStr</a> of  <a href="monoid_0.html#K7" target="_self">&lt;REAL,*&gt;</a>  st the <a href="struct_0.html#U1">carrier</a> of <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="txt" href="monoid_0.html#E9:119_1_1"><i><font color="Green">E57</font></i></a></i>;<br/>


</div>
