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<span class="kw">reconsider </span><font color="Maroon" title="c1">I</font> =  <a href="pre_poly.html#K2" title="PRE_POLY:func.2">&lt;*&gt;</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> as   <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c2">F</font> = <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><span class="p2"><a href="polyform.html#K3" title="POLYFORM:func.3">&lt;*</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span><a href="polyform.html#K3" title="POLYFORM:func.3">*&gt;</a></span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span> as   <a href="pre_poly.html#V1" title="PRE_POLY:attr.1">FinSequence-yielding</a>  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> ;<br/>
<span class="kw">take </span><font color="Maroon" title="c3">p</font> =  <a href="polyform.html#G1" title="POLYFORM:aggr.1">PolyhedronStr</a>(# <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c1">I</font> #); <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="c3">p</font> is  <a href="polyform.html#V2" title="POLYFORM:attr.2">polyhedron_1</a>  &amp; <font color="Maroon" title="c3">p</font> is  <a href="polyform.html#V3" title="POLYFORM:attr.3">polyhedron_2</a>  &amp; <font color="Maroon" title="c3">p</font> is  <a href="polyform.html#V4" title="POLYFORM:attr.4">polyhedron_3</a>  )</span><br/>

<a NAME="E3:33_1_1"/><span class="lab"><font color="Green" title="E27">A1</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">F</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/>
<a NAME="E4:33_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">I</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
;<br/>
<span class="kw">hence </span><a NAME="E5:33_1_1"/>
<font color="Maroon" title="c3">p</font> is  <a href="polyform.html#V2" title="POLYFORM:attr.2">polyhedron_1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:33_1_1"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="polyform.html#D1" target="_self" title="POLYFORM:def.1">Def1</a></span>; <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="c3">p</font> is  <a href="polyform.html#V3" title="POLYFORM:attr.3">polyhedron_2</a>  &amp; <font color="Maroon" title="c3">p</font> is  <a href="polyform.html#V4" title="POLYFORM:attr.4">polyhedron_3</a>  )</span><br/>

<a NAME="E6:33_1_1"/>
 for <font color="Olive" title="b1">n</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">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 holds <br/><font color="Maroon" title="c1">I</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> is   <a href="polyform.html#NM1" title="POLYFORM:NM.1">incidence-matrix</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span></span>)</span>
 
;<br/>
<span class="kw">hence </span><a NAME="E7:33_1_1"/>
<font color="Maroon" title="c3">p</font> is  <a href="polyform.html#V3" title="POLYFORM:attr.3">polyhedron_2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:33_1_1"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="polyform.html#D2" target="_self" title="POLYFORM:def.2">Def2</a></span>; <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="c3">p</font> is  <a href="polyform.html#V4" title="POLYFORM:attr.4">polyhedron_3</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c4">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="polyform.html#D3" target="_self" title="POLYFORM:def.3">POLYFORM:def 3</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="c4">n</font> &amp; <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the <a href="polyform.html#U1" title="POLYFORM:sel.1">PolytopsF</a> of <font color="Maroon" title="c3">p</font> implies (  not  the <a href="polyform.html#U1" title="POLYFORM:sel.1">PolytopsF</a> of <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp;  the <a href="polyform.html#U1" title="POLYFORM:sel.1">PolytopsF</a> of <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E8:33_1_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">n</font> &amp; <font color="Maroon" title="c4">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the <a href="polyform.html#U1" title="POLYFORM:sel.1">PolytopsF</a> of <font color="Maroon" title="c3">p</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not  the <a href="polyform.html#U1" title="POLYFORM:sel.1">PolytopsF</a> of <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp;  the <a href="polyform.html#U1" title="POLYFORM:sel.1">PolytopsF</a> of <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  )</span><br/>

<a NAME="E9:33_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:33_1_1"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:33_1_1"/>
(  not  the <a href="polyform.html#U1" title="POLYFORM:sel.1">PolytopsF</a> of <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp;  the <a href="polyform.html#U1" title="POLYFORM:sel.1">PolytopsF</a> of <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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