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

<span class="kw">let </span><font color="Maroon" title="c1">G1</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="group_1.html#V2" title="GROUP_1:attr.2">Group-like</a>   <a href="algstr_0.html#L3" title="ALGSTR_0:struct.3">multMagma</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <span class="p1">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <span class="p2"><a href="group_7.html#K5" title="GROUP_7:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c1">G1</font></span><a href="group_7.html#K5" title="GROUP_7:func.5">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="group_7.html#K10" title="GROUP_7:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G1</font></span>)</span></span><a href="group_7.html#K10" title="GROUP_7:func.10">*&gt;</a></span></span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">s</font> = <span class="p1"><a href="group_7.html#K10" title="GROUP_7:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G1</font></span>)</span></span><a href="group_7.html#K10" title="GROUP_7:func.10">*&gt;</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c3">f</font> = <span class="p1"><a href="group_7.html#K5" title="GROUP_7:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c1">G1</font></span><a href="group_7.html#K5" title="GROUP_7:func.5">*&gt;</a></span>;<br/>
<a NAME="E1:79"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="group_7.html#K10" title="GROUP_7:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G1</font></span>)</span></span><a href="group_7.html#K10" title="GROUP_7:func.10">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#D8" title="FINSEQ_1:def.8">FINSEQ_1:def 8</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">s</font> = <span class="p1"><a href="group_7.html#K10" title="GROUP_7:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G1</font></span>)</span></span><a href="group_7.html#K10" title="GROUP_7:func.10">*&gt;</a></span> as   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a>, <a class="ref" href="relat_1.html#D18" title="RELAT_1:def.18">RELAT_1:def 18</a></span>;<br/>
<a NAME="E3:79"/>
for <font color="Olive" title="b1">i</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> holds <br/> ex <font color="Olive" title="b2">G</font> being non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="group_1.html#V2" title="GROUP_1:attr.2">Group-like</a>   <a href="algstr_0.html#L3" title="ALGSTR_0:struct.3">multMagma</a>  st <br/>( <font color="Olive" title="b2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="group_7.html#K5" title="GROUP_7:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c1">G1</font></span><a href="group_7.html#K5" title="GROUP_7:func.5">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Olive" title="b2">G</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies  ex <font color="Olive" title="b1">G</font> being non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="group_1.html#V2" title="GROUP_1:attr.2">Group-like</a>   <a href="algstr_0.html#L3" title="ALGSTR_0:struct.3">multMagma</a>  st <br/>( <font color="Olive" title="b1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="group_7.html#K5" title="GROUP_7:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c1">G1</font></span><a href="group_7.html#K5" title="GROUP_7:func.5">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> &amp; <font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Olive" title="b1">G</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:79_1"/>
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default">1</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">G</font> being non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="group_1.html#V2" title="GROUP_1:attr.2">Group-like</a>   <a href="algstr_0.html#L3" title="ALGSTR_0:struct.3">multMagma</a>  st <br/>( <font color="Olive" title="b1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="group_7.html#K5" title="GROUP_7:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c1">G1</font></span><a href="group_7.html#K5" title="GROUP_7:func.5">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> &amp; <font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Olive" title="b1">G</font> )</span><br/>

<a NAME="E2:79_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A1</font></span>: 
<font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">G</font> = <span class="p1"><a href="group_7.html#K5" title="GROUP_7:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c1">G1</font></span><a href="group_7.html#K5" title="GROUP_7:func.5">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> as  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="group_1.html#V2" title="GROUP_1:attr.2">Group-like</a>   <a href="algstr_0.html#L3" title="ALGSTR_0:struct.3">multMagma</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D8" title="FINSEQ_1:def.8">FINSEQ_1:def 8</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c6">G</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="group_7.html#K5" title="GROUP_7:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c1">G1</font></span><a href="group_7.html#K5" title="GROUP_7:func.5">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> &amp; <font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c6">G</font> )</span><br/>

<a NAME="E4:79_1"/>
<span class="p1"><a href="group_7.html#K5" title="GROUP_7:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c1">G1</font></span><a href="group_7.html#K5" title="GROUP_7:func.5">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:79_1"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="ref" href="finseq_1.html#D8" title="FINSEQ_1:def.8">FINSEQ_1:def 8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:79_1"/>
( <font color="Maroon" title="c6">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="group_7.html#K5" title="GROUP_7:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c1">G1</font></span><a href="group_7.html#K5" title="GROUP_7:func.5">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> &amp; <font color="Maroon" title="c4">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c6">G</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:79_1"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="ref" href="finseq_1.html#D8" title="FINSEQ_1:def.8">FINSEQ_1:def 8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:79"/>
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <span class="p1">(<span class="default"><a href="group_7.html#K2" title="GROUP_7:func.2">product</a> <span class="p2"><a href="group_7.html#K5" title="GROUP_7:func.5">&lt;*</a><span class="default"><font color="Maroon" title="c1">G1</font></span><a href="group_7.html#K5" title="GROUP_7:func.5">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="group_7.html#K10" title="GROUP_7:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">G1</font></span>)</span></span><a href="group_7.html#K10" title="GROUP_7:func.10">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="group_7.html#T8" target="_self" title="GROUP_7:th.8">Th8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
