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<b>let </b><font color="Maroon" title="c1">G</font>, <font color="Maroon" title="c2">H</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="algstr_0.html#L3" title="ALGSTR_0:struct.3">multMagma</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <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#K11" title="GROUP_7:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font></span><a href="group_7.html#K11" title="GROUP_7:func.11">*&gt;</a></span></span>)</span>  ex <font color="Olive" title="b2">g</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> ex <font color="Olive" title="b3">h</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">H</font> st <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="group_7.html#K17" title="GROUP_7:func.17">&lt;*</a><span class="default"><font color="Olive" title="b2">g</font>,<font color="Olive" title="b3">h</font></span><a href="group_7.html#K17" title="GROUP_7:func.17">*&gt;</a></span></span><br/><b>let </b><font color="Maroon" title="c3">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <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#K11" title="GROUP_7:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font></span><a href="group_7.html#K11" title="GROUP_7:func.11">*&gt;</a></span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">g</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> ex <font color="Olive" title="b2">h</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">H</font> st <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="group_7.html#K17" title="GROUP_7:func.17">&lt;*</a><span class="default"><font color="Olive" title="b1">g</font>,<font color="Olive" title="b2">h</font></span><a href="group_7.html#K17" title="GROUP_7:func.17">*&gt;</a></span></span><br/>



<a NAME="E1:1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <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#K11" title="GROUP_7:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font></span><a href="group_7.html#K11" title="GROUP_7:func.11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2"><a href="group_7.html#K11" title="GROUP_7:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font></span><a href="group_7.html#K11" title="GROUP_7:func.11">*&gt;</a></span></span>)</span>
 
<b>by </b><i><a class="ref" href="group_7.html#D2" title="GROUP_7:def.2">GROUP_7:def 2</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">g</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E3:1"/><i><font color="Green" title="E4">A1</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">g</font>
 <b>and </b><br/><a NAME="E4:1"/><i><font color="Green" title="E5">A2</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2"><a href="group_7.html#K11" title="GROUP_7:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font></span><a href="group_7.html#K11" title="GROUP_7:func.11">*&gt;</a></span></span>)</span>
 <b>and </b><br/><a NAME="E5:1"/><i><font color="Green" title="E6">A3</font></i>: 
for <font color="Olive" title="b1">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2"><a href="group_7.html#K11" title="GROUP_7:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font></span><a href="group_7.html#K11" title="GROUP_7:func.11">*&gt;</a></span></span>)</span> holds <br/><font color="Maroon" title="c4">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2"><a href="group_7.html#K11" title="GROUP_7:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font></span><a href="group_7.html#K11" title="GROUP_7:func.11">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">y</font>
 <b>by </b><i><a class="ref" href="card_3.html#D5" title="CARD_3:def.5">CARD_3:def 5</a></i>;<br/>
<a NAME="E6:1"/><i><font color="Green" title="E7">A4</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2"><a href="group_7.html#K11" title="GROUP_7:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font></span><a href="group_7.html#K11" title="GROUP_7:func.11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">1,2</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<b>by </b><i><a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">g</font> = <font color="Maroon" title="c4">g</font> as   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> <b>by </b><i><a class="txt" href="topalg_4.html#E4:1"><i><font color="Green" title="E5">A2</font></i></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#D2" title="FINSEQ_1:def.2">FINSEQ_1:def 2</a></i>;<br/>
<a NAME="E8:1"/><i><font color="Green" title="E8">A5</font></i>: 
<font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2"><a href="group_7.html#K11" title="GROUP_7:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font></span><a href="group_7.html#K11" title="GROUP_7:func.11">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 
<b>by </b><i><a class="txt" href="topalg_4.html#E5:1"><i><font color="Green" title="E6">A3</font></i></a>, <a class="txt" href="topalg_4.html#E6:1"><i><font color="Green" title="E7">A4</font></i></a>, <a class="txt" href="topalg_4.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a></i>;<br/>
<a NAME="E9:1"/>
 ex <font color="Olive" title="b1">R</font> being   <a href="struct_0.html#L1" title="STRUCT_0:struct.1">1-sorted</a>  st <br/>( <font color="Olive" title="b1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="group_7.html#K11" title="GROUP_7:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font></span><a href="group_7.html#K11" title="GROUP_7:func.11">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp; <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2"><a href="group_7.html#K11" title="GROUP_7:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font></span><a href="group_7.html#K11" title="GROUP_7:func.11">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">R</font> )
 
<b>by </b><i><a class="txt" href="topalg_4.html#E1"><i><font color="Green" title="E1">Lm1</font></i></a>, <a class="ref" href="pralg_1.html#D13" title="PRALG_1:def.13">PRALG_1:def 13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">g1</font> = <font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="topalg_4.html#E8:1"><i><font color="Green" title="E8">A5</font></i></a>, <a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>;<br/>
<a NAME="E11:1"/><i><font color="Green" title="E9">A6</font></i>: 
<font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2"><a href="group_7.html#K11" title="GROUP_7:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font></span><a href="group_7.html#K11" title="GROUP_7:func.11">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
 
<b>by </b><i><a class="txt" href="topalg_4.html#E5:1"><i><font color="Green" title="E6">A3</font></i></a>, <a class="txt" href="topalg_4.html#E6:1"><i><font color="Green" title="E7">A4</font></i></a>, <a class="txt" href="topalg_4.html#E2"><i><font color="Green" title="E2">Lm2</font></i></a></i>;<br/>
<a NAME="E12:1"/>
 ex <font color="Olive" title="b1">R</font> being   <a href="struct_0.html#L1" title="STRUCT_0:struct.1">1-sorted</a>  st <br/>( <font color="Olive" title="b1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="group_7.html#K11" title="GROUP_7:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font></span><a href="group_7.html#K11" title="GROUP_7:func.11">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 &amp; <span class="p1">(<span class="default"><a href="pralg_1.html#K12" title="PRALG_1:func.12">Carrier</a> <span class="p2"><a href="group_7.html#K11" title="GROUP_7:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">H</font></span><a href="group_7.html#K11" title="GROUP_7:func.11">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">R</font> )
 
<b>by </b><i><a class="txt" href="topalg_4.html#E2"><i><font color="Green" title="E2">Lm2</font></i></a>, <a class="ref" href="pralg_1.html#D13" title="PRALG_1:def.13">PRALG_1:def 13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">h1</font> = <font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">H</font> <b>by </b><i><a class="txt" href="topalg_4.html#E11:1"><i><font color="Green" title="E9">A6</font></i></a>, <a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c6">g1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">h</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">H</font> st <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="group_7.html#K17" title="GROUP_7:func.17">&lt;*</a><span class="default"><font color="Maroon" title="c6">g1</font>,<font color="Olive" title="b1">h</font></span><a href="group_7.html#K17" title="GROUP_7:func.17">*&gt;</a></span></span><br/>

<b>take </b>
<font color="Maroon" title="c7">h1</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="group_7.html#K17" title="GROUP_7:func.17">&lt;*</a><span class="default"><font color="Maroon" title="c6">g1</font>,<font color="Maroon" title="c7">h1</font></span><a href="group_7.html#K17" title="GROUP_7:func.17">*&gt;</a></span></span><br/>

<a NAME="E14:1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<b>by </b><i><a class="txt" href="topalg_4.html#E4:1"><i><font color="Green" title="E5">A2</font></i></a>, <a class="txt" href="topalg_4.html#E6:1"><i><font color="Green" title="E7">A4</font></i></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>hence </b><a NAME="E15:1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="group_7.html#K17" title="GROUP_7:func.17">&lt;*</a><span class="default"><font color="Maroon" title="c6">g1</font>,<font color="Maroon" title="c7">h1</font></span><a href="group_7.html#K17" title="GROUP_7:func.17">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="topalg_4.html#E3:1"><i><font color="Green" title="E4">A1</font></i></a>, <a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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