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<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="algstr_0.html#L3" title="ALGSTR_0:struct.3">multMagma</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x1</font>, <font color="Olive" title="b2">x2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G1</font> holds  <span class="p1"><a href="group_7.html#K4" title="GROUP_7:func.4">&lt;*</a><span class="default"><font color="Olive" title="b1">x1</font></span><a href="group_7.html#K4" title="GROUP_7:func.4">*&gt;</a></span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1"><a href="group_7.html#K4" title="GROUP_7:func.4">&lt;*</a><span class="default"><font color="Olive" title="b2">x2</font></span><a href="group_7.html#K4" title="GROUP_7:func.4">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="group_7.html#K4" title="GROUP_7:func.4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">x1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">x2</font></span>)</span></span><a href="group_7.html#K4" title="GROUP_7:func.4">*&gt;</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">x1</font>, <font color="Maroon" title="c3">x2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="group_7.html#K4" title="GROUP_7:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c2">x1</font></span><a href="group_7.html#K4" title="GROUP_7:func.4">*&gt;</a></span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1"><a href="group_7.html#K4" title="GROUP_7:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">x2</font></span><a href="group_7.html#K4" title="GROUP_7:func.4">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="group_7.html#K4" title="GROUP_7:func.4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">x2</font></span>)</span></span><a href="group_7.html#K4" title="GROUP_7:func.4">*&gt;</a></span></span><br/>





<span class="kw">set </span><font color="Maroon" title="c4">G</font> = <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">G1</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>;<br/>
<a NAME="E1:64"/><span class="lab"><font color="Green" title="E28">A1</font></span>: 
<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">G1</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <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="ref" href="finseq_1.html#D8" title="FINSEQ_1:def.8">FINSEQ_1:def 8</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">l</font> = <span class="p1"><a href="group_7.html#K4" title="GROUP_7:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c2">x1</font></span><a href="group_7.html#K4" title="GROUP_7:func.4">*&gt;</a></span>, <font color="Maroon" title="c6">p</font> = <span class="p1"><a href="group_7.html#K4" title="GROUP_7:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">x2</font></span><a href="group_7.html#K4" title="GROUP_7:func.4">*&gt;</a></span>, <font color="Maroon" title="c7">lpl</font> = <span class="p1"><a href="group_7.html#K4" title="GROUP_7:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c2">x1</font></span><a href="group_7.html#K4" title="GROUP_7:func.4">*&gt;</a></span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1"><a href="group_7.html#K4" title="GROUP_7:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">x2</font></span><a href="group_7.html#K4" title="GROUP_7:func.4">*&gt;</a></span>, <font color="Maroon" title="c8">lpp</font> = <span class="p1"><a href="group_7.html#K4" title="GROUP_7:func.4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">x2</font></span>)</span></span><a href="group_7.html#K4" title="GROUP_7:func.4">*&gt;</a></span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <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="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">G1</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="group_7.html#D2" target="_self" title="GROUP_7:def.2">Def2</a></span>;<br/>
<a NAME="E3:64"/><span class="lab"><font color="Green" title="E29">A2</font></span>: 
<font color="Maroon" title="c5">l</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x1</font>
 
<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/>
<a NAME="E4:64"/><span class="lab"><font color="Green" title="E30">A3</font></span>: 
<font color="Maroon" title="c6">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x2</font>
 
<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/>
<a NAME="E5:64"/><span class="lab"><font color="Green" title="E31">A4</font></span>: 
<font color="Maroon" title="c8">lpp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">x2</font>
 
<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/>
<a NAME="E6:64"/><span class="lab"><font color="Green" title="E32">A5</font></span>: 
1 <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>
 
<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/>
<a NAME="E7:64"/><span class="lab"><font color="Green" title="E33">A6</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 holds <br/><font color="Maroon" title="c7">lpl</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">lpp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">k</font> &amp; <font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 implies <font color="Maroon" title="c7">lpl</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">lpp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:64_1"/><span class="lab"><font color="Green" title="E34">A7</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">k</font>
 <span class="kw">and </span><br/><a NAME="E2:64_1"/><span class="lab"><font color="Green" title="E35">A8</font></span>: 
<font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">lpl</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">lpp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font></span><br/>

<a NAME="E3:64_1"/>
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<a NAME="E4:64_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:64_1"><span class="lab"><font color="Green" title="E34">A7</font></span></a>, <a class="txt" href="#E2:64_1"><span class="lab"><font color="Green" title="E35">A8</font></span></a></span>;<br/>
<a NAME="E5:64_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T2" title="FINSEQ_1:th.2">FINSEQ_1:2</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:64_1"/>
<font color="Maroon" title="c7">lpl</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">lpp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:64"><span class="lab"><font color="Green" title="E32">A5</font></span></a>, <a class="txt" href="#E3:64"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="txt" href="#E4:64"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="txt" href="#E1:64"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="txt" href="#E5:64"><span class="lab"><font color="Green" title="E31">A4</font></span></a>, <a class="ref" href="group_7.html#T1" target="_self" title="GROUP_7:th.1">Th4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">lpl</font> = 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c1">G1</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#T9" title="CARD_3:th.9">CARD_3:9</a></span>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T2" title="FINSEQ_1:th.2">FINSEQ_1:2</a>, <a class="ref" href="partfun1.html#D2" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>
;<br/>
<a NAME="E9:64"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A9</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">lpl</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E10:64"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">lpp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T39" title="FINSEQ_1:th.39">FINSEQ_1:39</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:64"/>
<span class="p1"><a href="group_7.html#K4" title="GROUP_7:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c2">x1</font></span><a href="group_7.html#K4" title="GROUP_7:func.4">*&gt;</a></span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1"><a href="group_7.html#K4" title="GROUP_7:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">x2</font></span><a href="group_7.html#K4" title="GROUP_7:func.4">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="group_7.html#K4" title="GROUP_7:func.4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">x1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c3">x2</font></span>)</span></span><a href="group_7.html#K4" title="GROUP_7:func.4">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:64"><span class="lab"><font color="Green" title="E34">A9</font></span></a>, <a class="txt" href="#E7:64"><span class="lab"><font color="Green" title="E33">A6</font></span></a>, <a class="ref" href="finseq_1.html#T14" title="FINSEQ_1:th.14">FINSEQ_1:14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
