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<span class="kw">let </span><font color="Maroon" title="c1">G1</font>, <font color="Maroon" title="c2">G2</font>, <font color="Maroon" title="c3">G3</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="Red">::  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><br/> for <font color="Olive" title="b3">y1</font>, <font color="Olive" title="b4">y2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G2</font><br/> for <font color="Olive" title="b5">z1</font>, <font color="Olive" title="b6">z2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">G3</font> holds  <span class="p1"><a href="group_7.html#K24" title="GROUP_7:func.24">&lt;*</a><span class="default"><font color="Olive" title="b1">x1</font>,<font color="Olive" title="b3">y1</font>,<font color="Olive" title="b5">z1</font></span><a href="group_7.html#K24" title="GROUP_7:func.24">*&gt;</a></span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1"><a href="group_7.html#K24" title="GROUP_7:func.24">&lt;*</a><span class="default"><font color="Olive" title="b2">x2</font>,<font color="Olive" title="b4">y2</font>,<font color="Olive" title="b6">z2</font></span><a href="group_7.html#K24" title="GROUP_7:func.24">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="group_7.html#K24" title="GROUP_7:func.24">&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 class="p2">(<span class="default"><font color="Olive" title="b3">y1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b4">y2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b5">z1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b6">z2</font></span>)</span></span><a href="group_7.html#K24" title="GROUP_7:func.24">*&gt;</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">x1</font>, <font color="Maroon" title="c5">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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">y1</font>, <font color="Olive" title="b2">y2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G2</font><br/> for <font color="Olive" title="b3">z1</font>, <font color="Olive" title="b4">z2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">G3</font> holds  <span class="p1"><a href="group_7.html#K24" title="GROUP_7:func.24">&lt;*</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Olive" title="b1">y1</font>,<font color="Olive" title="b3">z1</font></span><a href="group_7.html#K24" title="GROUP_7:func.24">*&gt;</a></span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1"><a href="group_7.html#K24" title="GROUP_7:func.24">&lt;*</a><span class="default"><font color="Maroon" title="c5">x2</font>,<font color="Olive" title="b2">y2</font>,<font color="Olive" title="b4">z2</font></span><a href="group_7.html#K24" title="GROUP_7:func.24">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="group_7.html#K24" title="GROUP_7:func.24">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">x2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">y1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">y2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b3">z1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b4">z2</font></span>)</span></span><a href="group_7.html#K24" title="GROUP_7:func.24">*&gt;</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c6">y1</font>, <font color="Maroon" title="c7">y2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">G2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">z1</font>, <font color="Olive" title="b2">z2</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">G3</font> holds  <span class="p1"><a href="group_7.html#K24" title="GROUP_7:func.24">&lt;*</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c6">y1</font>,<font color="Olive" title="b1">z1</font></span><a href="group_7.html#K24" title="GROUP_7:func.24">*&gt;</a></span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1"><a href="group_7.html#K24" title="GROUP_7:func.24">&lt;*</a><span class="default"><font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c7">y2</font>,<font color="Olive" title="b2">z2</font></span><a href="group_7.html#K24" title="GROUP_7:func.24">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="group_7.html#K24" title="GROUP_7:func.24">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">x2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">y1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">y2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Olive" title="b1">z1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">z2</font></span>)</span></span><a href="group_7.html#K24" title="GROUP_7:func.24">*&gt;</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c8">z1</font>, <font color="Maroon" title="c9">z2</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">G3</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="group_7.html#K24" title="GROUP_7:func.24">&lt;*</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c6">y1</font>,<font color="Maroon" title="c8">z1</font></span><a href="group_7.html#K24" title="GROUP_7:func.24">*&gt;</a></span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1"><a href="group_7.html#K24" title="GROUP_7:func.24">&lt;*</a><span class="default"><font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c7">y2</font>,<font color="Maroon" title="c9">z2</font></span><a href="group_7.html#K24" title="GROUP_7:func.24">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="group_7.html#K24" title="GROUP_7:func.24">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">x2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">y1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">y2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">z1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">z2</font></span>)</span></span><a href="group_7.html#K24" title="GROUP_7:func.24">*&gt;</a></span></span><br/>

















<span class="kw">set </span><font color="Maroon" title="c10">G</font> = <span class="p1"><a href="group_7.html#K18" title="GROUP_7:func.18">&lt;*</a><span class="default"><font color="Maroon" title="c1">G1</font>,<font color="Maroon" title="c2">G2</font>,<font color="Maroon" title="c3">G3</font></span><a href="group_7.html#K18" title="GROUP_7:func.18">*&gt;</a></span>;<br/>
<a NAME="E1:78"/><span class="lab"><font color="Green" title="E34">A1</font></span>: 
3 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default">1,2,3</span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></span>;<br/>
<a NAME="E2:78"/><span class="lab"><font color="Green" title="E35">A2</font></span>: 
<span class="p1"><a href="group_7.html#K18" title="GROUP_7:func.18">&lt;*</a><span class="default"><font color="Maroon" title="c1">G1</font>,<font color="Maroon" title="c2">G2</font>,<font color="Maroon" title="c3">G3</font></span><a href="group_7.html#K18" title="GROUP_7:func.18">*&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#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></span>;<br/>
<a NAME="E3:78"/><span class="lab"><font color="Green" title="E36">A3</font></span>: 
<span class="p1"><a href="group_7.html#K18" title="GROUP_7:func.18">&lt;*</a><span class="default"><font color="Maroon" title="c1">G1</font>,<font color="Maroon" title="c2">G2</font>,<font color="Maroon" title="c3">G3</font></span><a href="group_7.html#K18" title="GROUP_7:func.18">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">G3</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></span>;<br/>
<a NAME="E4:78"/><span class="lab"><font color="Green" title="E37">A4</font></span>: 
<span class="p1"><a href="group_7.html#K18" title="GROUP_7:func.18">&lt;*</a><span class="default"><font color="Maroon" title="c1">G1</font>,<font color="Maroon" title="c2">G2</font>,<font color="Maroon" title="c3">G3</font></span><a href="group_7.html#K18" title="GROUP_7:func.18">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">l</font> = <span class="p1"><a href="group_7.html#K24" title="GROUP_7:func.24">&lt;*</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c6">y1</font>,<font color="Maroon" title="c8">z1</font></span><a href="group_7.html#K24" title="GROUP_7:func.24">*&gt;</a></span>, <font color="Maroon" title="c12">p</font> = <span class="p1"><a href="group_7.html#K24" title="GROUP_7:func.24">&lt;*</a><span class="default"><font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c7">y2</font>,<font color="Maroon" title="c9">z2</font></span><a href="group_7.html#K24" title="GROUP_7:func.24">*&gt;</a></span>, <font color="Maroon" title="c13">lpl</font> = <span class="p1"><a href="group_7.html#K24" title="GROUP_7:func.24">&lt;*</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c6">y1</font>,<font color="Maroon" title="c8">z1</font></span><a href="group_7.html#K24" title="GROUP_7:func.24">*&gt;</a></span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1"><a href="group_7.html#K24" title="GROUP_7:func.24">&lt;*</a><span class="default"><font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c7">y2</font>,<font color="Maroon" title="c9">z2</font></span><a href="group_7.html#K24" title="GROUP_7:func.24">*&gt;</a></span>, <font color="Maroon" title="c14">lpp</font> = <span class="p1"><a href="group_7.html#K24" title="GROUP_7:func.24">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">x2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">y1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">y2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">z1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">z2</font></span>)</span></span><a href="group_7.html#K24" title="GROUP_7:func.24">*&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="group_7.html#K18" title="GROUP_7:func.18">&lt;*</a><span class="default"><font color="Maroon" title="c1">G1</font>,<font color="Maroon" title="c2">G2</font>,<font color="Maroon" title="c3">G3</font></span><a href="group_7.html#K18" title="GROUP_7:func.18">*&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="E6:78"/><span class="lab"><font color="Green" title="E38">A5</font></span>: 
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default">1,2,3</span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></span>;<br/>
<a NAME="E7:78"/><span class="lab"><font color="Green" title="E39">A6</font></span>: 
<font color="Maroon" title="c11">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="c4">x1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></span>;<br/>
<a NAME="E8:78"/><span class="lab"><font color="Green" title="E40">A7</font></span>: 
<font color="Maroon" title="c11">l</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">z1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></span>;<br/>
<a NAME="E9:78"/><span class="lab"><font color="Green" title="E41">A8</font></span>: 
<font color="Maroon" title="c11">l</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></span>;<br/>
<a NAME="E10:78"/><span class="lab"><font color="Green" title="E42">A9</font></span>: 
<font color="Maroon" title="c14">lpp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">z1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">z2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></span>;<br/>
<a NAME="E11:78"/><span class="lab"><font color="Green" title="E43">A10</font></span>: 
<font color="Maroon" title="c14">lpp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">y2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></span>;<br/>
<a NAME="E12:78"/><span class="lab"><font color="Green" title="E44">A11</font></span>: 
<font color="Maroon" title="c14">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="c4">x1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">x2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></span>;<br/>
<a NAME="E13:78"/><span class="lab"><font color="Green" title="E45">A12</font></span>: 
<font color="Maroon" title="c12">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">z2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></span>;<br/>
<a NAME="E14:78"/><span class="lab"><font color="Green" title="E46">A13</font></span>: 
<font color="Maroon" title="c12">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></span>;<br/>
<a NAME="E15:78"/><span class="lab"><font color="Green" title="E47">A14</font></span>: 
<font color="Maroon" title="c12">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="c5">x2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></span>;<br/>
<a NAME="E16:78"/><span class="lab"><font color="Green" title="E48">A15</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default">1,2,3</span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></span>;<br/>
<a NAME="E17:78"/><span class="lab"><font color="Green" title="E49">A16</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> 3 holds <br/><font color="Maroon" title="c13">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="c14">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="78_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">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="Red">::  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="c15">k</font> &amp; <font color="Maroon" title="c15">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3 implies <font color="Maroon" title="c13">lpl</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">lpp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">k</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:78_1"/><span class="lab"><font color="Green" title="E50">A17</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">k</font>
 <span class="kw">and </span><br/><a NAME="E2:78_1"/><span class="lab"><font color="Green" title="E51">A18</font></span>: 
<font color="Maroon" title="c15">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 3
 ; <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="c13">lpl</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">lpp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">k</font></span><br/>

<a NAME="E3:78_1"/>
<font color="Maroon" title="c15">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#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<a NAME="E4:78_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A19</font></span>: 
<font color="Maroon" title="c15">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> 3
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:78_1"><span class="lab"><font color="Green" title="E50">A17</font></span></a>, <a class="txt" href="#E2:78_1"><span class="lab"><font color="Green" title="E51">A18</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:78_1_1"/>
( <font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:78_1"><span class="lab"><font color="Green" title="E52">A19</font></span></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a>, <a class="ref" href="finseq_3.html#T1" title="FINSEQ_3:th.1">FINSEQ_3:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:78_1_1_1"/>
<font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <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="c13">lpl</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">lpp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">k</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:78_1_1_1"/>
<font color="Maroon" title="c13">lpl</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">lpp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:78"><span class="lab"><font color="Green" title="E48">A15</font></span></a>, <a class="txt" href="#E7:78"><span class="lab"><font color="Green" title="E39">A6</font></span></a>, <a class="txt" href="#E15:78"><span class="lab"><font color="Green" title="E47">A14</font></span></a>, <a class="txt" href="#E2:78"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="txt" href="#E12:78"><span class="lab"><font color="Green" title="E44">A11</font></span></a>, <a class="ref" href="group_7.html#T4" target="_self" title="GROUP_7:th.4">Th4</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:78_1_1_2"/>
<font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ; <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="c13">lpl</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">lpp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">k</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:78_1_1_2"/>
<font color="Maroon" title="c13">lpl</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">lpp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:78"><span class="lab"><font color="Green" title="E38">A5</font></span></a>, <a class="txt" href="#E9:78"><span class="lab"><font color="Green" title="E41">A8</font></span></a>, <a class="txt" href="#E14:78"><span class="lab"><font color="Green" title="E46">A13</font></span></a>, <a class="txt" href="#E4:78"><span class="lab"><font color="Green" title="E37">A4</font></span></a>, <a class="txt" href="#E11:78"><span class="lab"><font color="Green" title="E43">A10</font></span></a>, <a class="ref" href="group_7.html#T4" target="_self" title="GROUP_7:th.4">Th4</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="78_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:78_1_1_3"/>
<font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 ; <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="c13">lpl</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">lpp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">k</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:78_1_1_3"/>
<font color="Maroon" title="c13">lpl</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">lpp</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:78"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="txt" href="#E8:78"><span class="lab"><font color="Green" title="E40">A7</font></span></a>, <a class="txt" href="#E13:78"><span class="lab"><font color="Green" title="E45">A12</font></span></a>, <a class="txt" href="#E3:78"><span class="lab"><font color="Green" title="E36">A3</font></span></a>, <a class="txt" href="#E10:78"><span class="lab"><font color="Green" title="E42">A9</font></span></a>, <a class="ref" href="group_7.html#T4" target="_self" title="GROUP_7:th.4">Th4</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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c13">lpl</font> = 
 <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#K18" title="GROUP_7:func.18">&lt;*</a><span class="default"><font color="Maroon" title="c1">G1</font>,<font color="Maroon" title="c2">G2</font>,<font color="Maroon" title="c3">G3</font></span><a href="group_7.html#K18" title="GROUP_7:func.18">*&gt;</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></span>
<br/>.= 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 3
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T1" title="FINSEQ_3:th.1">FINSEQ_3:1</a>, <a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>
;<br/>
<a NAME="E19:78"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A20</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c13">lpl</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 
<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="E20:78"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">lpp</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T62" title="FINSEQ_1:th.62">FINSEQ_1:62</a></span>;<br/>
<span class="kw">hence </span><a NAME="E21:78"/>
<span class="p1"><a href="group_7.html#K24" title="GROUP_7:func.24">&lt;*</a><span class="default"><font color="Maroon" title="c4">x1</font>,<font color="Maroon" title="c6">y1</font>,<font color="Maroon" title="c8">z1</font></span><a href="group_7.html#K24" title="GROUP_7:func.24">*&gt;</a></span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1"><a href="group_7.html#K24" title="GROUP_7:func.24">&lt;*</a><span class="default"><font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c7">y2</font>,<font color="Maroon" title="c9">z2</font></span><a href="group_7.html#K24" title="GROUP_7:func.24">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="group_7.html#K24" title="GROUP_7:func.24">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c5">x2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">y1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">y2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">z1</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">z2</font></span>)</span></span><a href="group_7.html#K24" title="GROUP_7:func.24">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:78"><span class="lab"><font color="Green" title="E50">A20</font></span></a>, <a class="txt" href="#E17:78"><span class="lab"><font color="Green" title="E49">A16</font></span></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</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>
