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<a NAME="E1:72_1_1"/>
 <a href="matrix15.html#K5" title="MATRIX15:func.5">Solutions_of</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="matrix13.html#K13" title="MATRIX13:func.13">-VectSp_over</a> <font color="Maroon" title="c1">K</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix15.html#K5" title="MATRIX15:func.5">Solutions_of</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">b</font> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="matrix13.html#K13" title="MATRIX13:func.13">-VectSp_over</a> <font color="Maroon" title="c1">K</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:72_1_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix15.html#K5" title="MATRIX15:func.5">Solutions_of</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">b</font>
 ; <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="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="matrix13.html#K13" title="MATRIX13:func.13">-VectSp_over</a> <font color="Maroon" title="c1">K</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">f</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c1">K</font><span class="kw"> such that </span><br/><a NAME="E3:72_1_1_1"/><span class="lab"><font color="Green" title="E57">A1</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">f</font>
 <span class="kw">and </span><br/><a NAME="E4:72_1_1_1"/><span class="lab"><font color="Green" title="E58">A2</font></span>: 
 <a href="matrix15.html#K3" title="MATRIX15:func.3">ColVec2Mx</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix15.html#K4" title="MATRIX15:func.4">Solutions_of</a> <font color="Maroon" title="c2">A</font>,<span class="p1">(<span class="default"><a href="matrix15.html#K3" title="MATRIX15:func.3">ColVec2Mx</a> <font color="Maroon" title="c3">b</font></span>)</span>
 ;<br/>
<a NAME="E5:72_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:72_1_1_1"><span class="lab"><font color="Green" title="E58">A2</font></span></a>, <a class="ref" href="matrix15.html#T59" target="_self" title="MATRIX15:th.59">Th59</a></span>;<br/>
<a NAME="E6:72_1_1_1"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="matrix13.html#K13" title="MATRIX13:func.13">-VectSp_over</a> <font color="Maroon" title="c1">K</font></span>)</span> &amp; <font color="Maroon" title="c5">f</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">K</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T110" title="FINSEQ_2:th.110">FINSEQ_2:110</a>, <a class="ref" href="matrix13.html#T102" title="MATRIX13:th.102">MATRIX13:102</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:72_1_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="matrix13.html#K13" title="MATRIX13:func.13">-VectSp_over</a> <font color="Maroon" title="c1">K</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:72_1_1_1"><span class="lab"><font color="Green" title="E57">A1</font></span></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>
<span class="kw">hence </span><a NAME="E2:72_1_1"/>
 <a href="matrix15.html#K5" title="MATRIX15:func.5">Solutions_of</a> <font color="Maroon" title="c2">A</font>,<font color="Maroon" title="c3">b</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="matrix13.html#K13" title="MATRIX13:func.13">-VectSp_over</a> <font color="Maroon" title="c1">K</font></span>)</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/>


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