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<b>let </b><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">m</font>, <font color="Maroon" title="c3">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</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">K</font> being  <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a><br/> for <font color="Olive" title="b2">A</font> being  <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>,<font color="Olive" title="b1">K</font><br/> for <font color="Olive" title="b3">B</font> being  <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">k</font>,<font color="Olive" title="b1">K</font><br/> for <font color="Olive" title="b4">P</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>, <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> holds <br/> (  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> <font color="Olive" title="b2">A</font>,<font color="Olive" title="b3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">A</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Olive" title="b4">P</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b3">B</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Olive" title="b4">P</font></span>)</span> &amp; ( <font color="Olive" title="b4">P</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> implies  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> <font color="Olive" title="b2">A</font>,<font color="Olive" title="b3">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">A</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Olive" title="b4">P</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b3">B</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Olive" title="b4">P</font></span>)</span> ) )</span><br/><b>let </b><font color="Maroon" title="c4">K</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</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">A</font> being  <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c4">K</font><br/> for <font color="Olive" title="b2">B</font> being  <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">K</font><br/> for <font color="Olive" title="b3">P</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>, <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> holds <br/> (  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Olive" title="b3">P</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b2">B</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Olive" title="b3">P</font></span>)</span> &amp; ( <font color="Olive" title="b3">P</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> implies  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> <font color="Olive" title="b1">A</font>,<font color="Olive" title="b2">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">A</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Olive" title="b3">P</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b2">B</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Olive" title="b3">P</font></span>)</span> ) )</span><br/><b>let </b><font color="Maroon" title="c5">A</font> be   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c4">K</font>; <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">B</font> being  <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">K</font><br/> for <font color="Olive" title="b2">P</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>, <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> holds <br/> (  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> <font color="Maroon" title="c5">A</font>,<font color="Olive" title="b1">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Olive" title="b2">P</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">B</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Olive" title="b2">P</font></span>)</span> &amp; ( <font color="Olive" title="b2">P</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> implies  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> <font color="Maroon" title="c5">A</font>,<font color="Olive" title="b1">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Olive" title="b2">P</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">B</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Olive" title="b2">P</font></span>)</span> ) )</span><br/><b>let </b><font color="Maroon" title="c6">B</font> be   <a href="matrix_1.html#NM6" title="MATRIX_1:NM.6">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">K</font>; <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">P</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>, <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> holds <br/> (  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Olive" title="b1">P</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Olive" title="b1">P</font></span>)</span> &amp; ( <font color="Olive" title="b1">P</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> implies  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Olive" title="b1">P</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Olive" title="b1">P</font></span>)</span> ) )</span><br/><b>let </b><font color="Maroon" title="c7">P</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>, <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Maroon" title="c7">P</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Maroon" title="c7">P</font></span>)</span> &amp; ( <font color="Maroon" title="c7">P</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> implies  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Maroon" title="c7">P</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Maroon" title="c7">P</font></span>)</span> ) )</span><br/>













<b>set </b><font color="Maroon" title="c8">IDn</font> =  <a href="finseq_2.html#K1" title="FINSEQ_2:func.1">idseq</a> <font color="Maroon" title="c1">n</font>;<br/>
<a NAME="E1:57"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_2.html#K1" title="FINSEQ_2:func.1">idseq</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="finseq_2.html#K1" title="FINSEQ_2:func.1">idseq</a> <font color="Maroon" title="c1">n</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="finseq_2.html#T52" title="FINSEQ_2:th.52">FINSEQ_2:52</a>, <a class="ref" href="finseq_2.html#T55" title="FINSEQ_2:th.55">FINSEQ_2:55</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c9">IDn</font> =  <a href="finseq_2.html#K1" title="FINSEQ_2:func.1">idseq</a> <font color="Maroon" title="c1">n</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T110" title="FINSEQ_2:th.110">FINSEQ_2:110</a></i>;<br/>
<a NAME="E3:57"/><i><font color="Green" title="E47">A1</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">IDn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#T67" title="FUNCT_2:th.67">FUNCT_2:67</a>, <a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c10">IDnP</font> = <font color="Maroon" title="c9">IDn</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">P</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T51" title="FINSEQ_2:th.51">FINSEQ_2:51</a></i>;<br/>
<a NAME="E5:57"/><i><font color="Green" title="E48">A2</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E6:57"/>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">IDnP</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="#E3:57"><i><font color="Green" title="E47">A1</font></i></a>, <a class="ref" href="relat_1.html#T79" title="RELAT_1:th.79">RELAT_1:79</a></i>;<br/>
<a NAME="E7:57"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">IDnP</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="#E5:57"><i><font color="Green" title="E48">A2</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c11">IDnP</font> = <font color="Maroon" title="c10">IDnP</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c1">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_2.html#T110" title="FINSEQ_2:th.110">FINSEQ_2:110</a></i>;<br/>
<a NAME="E9:57"/><i><font color="Green" title="E49">A3</font></i>: 
( <font color="Maroon" title="c9">IDn</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span> &amp; <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">A</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; <span class="p1">(<span class="default"><a href="finseq_2.html#K1" title="FINSEQ_2:func.1">idseq</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c7">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">P</font> )
 
<b>by </b><i><a class="txt" href="#E3:57"><i><font color="Green" title="E47">A1</font></i></a>, <a class="ref" href="finseq_1.html#T78" title="FINSEQ_1:th.78">FINSEQ_1:78</a>, <a class="ref" href="finseq_3.html#T54" title="FINSEQ_3:th.54">FINSEQ_3:54</a>, <a class="ref" href="matrix_1.html#D3" title="MATRIX_1:def.3">MATRIX_1:def 3</a>, <a class="ref" href="relat_1.html#T79" title="RELAT_1:th.79">RELAT_1:79</a></i>;<br/>
<b>then </b><i><font color="Green" title="E50">A4</font></i>:  <a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c11">IDnP</font>,<span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c5">A</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">A</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">P</font>
<b>by </b><i><a class="ref" href="matrix13.html#T33" title="MATRIX13:th.33">MATRIX13:33</a></i>
<br/>.= 
<font color="Maroon" title="c5">A</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Maroon" title="c7">P</font>
<b>by </b><i><a class="ref" href="matrix13.html#T46" title="MATRIX13:th.46">MATRIX13:46</a></i>
;<br/>
<i><font color="Green" title="E51">A5</font></i>:  <a href="matrix13.html#K1" title="MATRIX13:func.1">Segm</a> <font color="Maroon" title="c6">B</font>,<font color="Maroon" title="c11">IDnP</font>,<span class="p1">(<span class="default"><a href="matrix13.html#K5" title="MATRIX13:func.5">Sgm</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="matrix13.html#K6" title="MATRIX13:func.6">Segm</a> <font color="Maroon" title="c6">B</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c6">B</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">P</font>
<b>by </b><i><a class="txt" href="#E9:57"><i><font color="Green" title="E49">A3</font></i></a>, <a class="ref" href="matrix13.html#T33" title="MATRIX13:th.33">MATRIX13:33</a></i>
<br/>.= 
<font color="Maroon" title="c6">B</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Maroon" title="c7">P</font>
<b>by </b><i><a class="ref" href="matrix13.html#T46" title="MATRIX13:th.46">MATRIX13:46</a></i>
;<br/>
<a NAME="E12:57"/><i><font color="Green" title="E52">A6</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">IDnP</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">A</font>
 
<b>by </b><i><a class="txt" href="#E3:57"><i><font color="Green" title="E47">A1</font></i></a>, <a class="txt" href="#E9:57"><i><font color="Green" title="E49">A3</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:57_3"/>
( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  or <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_3_1"><b>suppose </b></a><a NAME="E1:57_3_1"/><i><font color="Green" title="E53">A7</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Maroon" title="c7">P</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Maroon" title="c7">P</font></span>)</span> &amp; ( <font color="Maroon" title="c7">P</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> implies  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Maroon" title="c7">P</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Maroon" title="c7">P</font></span>)</span> ) )</span><br/><div class="add"><b>hence </b><a NAME="E2:57_3_1"/>
 <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Maroon" title="c7">P</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Maroon" title="c7">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E10:57"><i><font color="Green" title="E50">A4</font></i></a>, <a class="txt" href="#E11:57"><i><font color="Green" title="E51">A5</font></i></a>, <a class="txt" href="#E12:57"><i><font color="Green" title="E52">A6</font></i></a>, <a class="ref" href="matrix15.html#T42" target="_self" title="MATRIX15:th.42">Th42</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c7">P</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> implies  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Maroon" title="c7">P</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Maroon" title="c7">P</font></span>)</span> )</span><br/><b>assume </b><a NAME="E3:57_3_1"/><i><font color="Green" title="E54">A8</font></i>: 
<font color="Maroon" title="c7">P</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Maroon" title="c7">P</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Maroon" title="c7">P</font></span>)</span></span><br/><a NAME="E4:57_3_1"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span>
 ;<br/><a NAME="E5:57_3_1"/><b>then </b>
<font color="Maroon" title="c7">P</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>
 <b>by </b><i><a class="txt" href="#E3:57_3_1"><i><font color="Green" title="E54">A8</font></i></a>, <a class="ref" href="stirl2_1.html#T70" title="STIRL2_1:th.70">STIRL2_1:70</a></i>;<br/><a NAME="E6:57_3_1"/><b>then </b><i><font color="Green" title="E55">A9</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a></i>;<br/><a NAME="E7:57_3_1"/><i><font color="Green" title="E56">A10</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="txt" href="#E9:57"><i><font color="Green" title="E49">A3</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E8:57_3_1"/><b>then </b>
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">IDnP</font></span>)</span> holds <br/>(  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">A</font>,<font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">A</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">K</font></span>)</span> &amp;  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c6">B</font>,<font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c6">B</font></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c4">K</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E9:57"><i><font color="Green" title="E49">A3</font></i></a>, <a class="txt" href="#E6:57_3_1"><i><font color="Green" title="E55">A9</font></i></a>, <a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a></i>;<br/><b>hence </b><a NAME="E9:57_3_1"/>
 <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Maroon" title="c7">P</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Maroon" title="c7">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E10:57"><i><font color="Green" title="E50">A4</font></i></a>, <a class="txt" href="#E11:57"><i><font color="Green" title="E51">A5</font></i></a>, <a class="ref" href="matrix15.html#T43" target="_self" title="MATRIX15:th.43">Th43</a>, <a class="txt" href="#E3:57"><i><font color="Green" title="E47">A1</font></i></a>, <a class="txt" href="#E9:57"><i><font color="Green" title="E49">A3</font></i></a>, <a class="txt" href="#E1:57_3_1"><i><font color="Green" title="E53">A7</font></i></a>, <a class="txt" href="#E7:57_3_1"><i><font color="Green" title="E56">A10</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="57_3_2"><b>suppose </b></a><a NAME="E1:57_3_2"/>
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Maroon" title="c7">P</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Maroon" title="c7">P</font></span>)</span> &amp; ( <font color="Maroon" title="c7">P</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> implies  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Maroon" title="c7">P</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Maroon" title="c7">P</font></span>)</span> ) )</span><br/><div class="add"><a NAME="E2:57_3_2"/><b>then </b>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Maroon" title="c7">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Maroon" title="c7">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  )
 <b>by </b><i><a class="ref" href="matrix_1.html#T23" title="MATRIX_1:th.23">MATRIX_1:23</a></i>;<br/><a NAME="E3:57_3_2"/><b>then </b>
( <font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c5">A</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Maroon" title="c7">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c6">B</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Maroon" title="c7">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 ;<br/><b>hence </b><a NAME="E4:57_3_2"/>
(  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Maroon" title="c7">P</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Maroon" title="c7">P</font></span>)</span> &amp; ( <font color="Maroon" title="c7">P</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> implies  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> <font color="Maroon" title="c5">A</font>,<font color="Maroon" title="c6">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">A</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Maroon" title="c7">P</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">B</font> <a href="matrix11.html#K4" title="MATRIX11:func.4">*</a> <font color="Maroon" title="c7">P</font></span>)</span> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

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