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<span class="kw">let </span><font color="Maroon" title="c1">n</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"> 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">a9</font>, <font color="Olive" title="b3">b9</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">K</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c1">n</font> <a href="matrix_2.html#K2" title="MATRIX_2:func.2">--&gt;</a> <font color="Olive" title="b2">a9</font></span>)</span> <a href="matrix_1.html#K14" title="MATRIX_1:func.14">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c1">n</font> <a href="matrix_2.html#K2" title="MATRIX_2:func.2">--&gt;</a> <font color="Olive" title="b3">b9</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c1">n</font> <a href="matrix_2.html#K2" title="MATRIX_2:func.2">--&gt;</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">a9</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b3">b9</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">K</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</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">a9</font>, <font color="Olive" title="b2">b9</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">K</font> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c1">n</font> <a href="matrix_2.html#K2" title="MATRIX_2:func.2">--&gt;</a> <font color="Olive" title="b1">a9</font></span>)</span> <a href="matrix_1.html#K14" title="MATRIX_1:func.14">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c1">n</font> <a href="matrix_2.html#K2" title="MATRIX_2:func.2">--&gt;</a> <font color="Olive" title="b2">b9</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c1">n</font> <a href="matrix_2.html#K2" title="MATRIX_2:func.2">--&gt;</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a9</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Olive" title="b2">b9</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">a9</font>, <font color="Maroon" title="c4">b9</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">K</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">(<span class="default"><font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c1">n</font> <a href="matrix_2.html#K2" title="MATRIX_2:func.2">--&gt;</a> <font color="Maroon" title="c3">a9</font></span>)</span> <a href="matrix_1.html#K14" title="MATRIX_1:func.14">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c1">n</font> <a href="matrix_2.html#K2" title="MATRIX_2:func.2">--&gt;</a> <font color="Maroon" title="c4">b9</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c1">n</font> <a href="matrix_2.html#K2" title="MATRIX_2:func.2">--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a9</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">b9</font></span>)</span></span><br/>







<a NAME="E1:4"/><span class="lab"><font color="Green" title="E2">A1</font></span>: 
 <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c1">n</font> <a href="matrix_2.html#K2" title="MATRIX_2:func.2">--&gt;</a> <font color="Maroon" title="c3">a9</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c1">n</font> <a href="matrix_2.html#K2" title="MATRIX_2:func.2">--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a9</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">b9</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T27" title="MATRIX_1:th.27">MATRIX_1:27</a></span>;<br/>
<a NAME="E2:4"/><span class="lab"><font color="Green" title="E3">A2</font></span>: 
 <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c1">n</font> <a href="matrix_2.html#K2" title="MATRIX_2:func.2">--&gt;</a> <font color="Maroon" title="c3">a9</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c1">n</font> <a href="matrix_2.html#K2" title="MATRIX_2:func.2">--&gt;</a> <font color="Maroon" title="c4">b9</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T27" title="MATRIX_1:th.27">MATRIX_1:27</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">i</font>, <font color="Maroon" title="c6">j</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"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c1">n</font> <a href="matrix_2.html#K2" title="MATRIX_2:func.2">--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a9</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">b9</font></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c1">n</font> <a href="matrix_2.html#K2" title="MATRIX_2:func.2">--&gt;</a> <font color="Maroon" title="c3">a9</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c1">n</font> <a href="matrix_2.html#K2" title="MATRIX_2:func.2">--&gt;</a> <font color="Maroon" title="c4">b9</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c1">n</font> <a href="matrix_2.html#K2" title="MATRIX_2:func.2">--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a9</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">b9</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font> )</span><br/><span class="kw">assume </span><a NAME="E1:4_1"/><span class="lab"><font color="Green" title="E4">A3</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c1">n</font> <a href="matrix_2.html#K2" title="MATRIX_2:func.2">--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a9</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">b9</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c1">n</font> <a href="matrix_2.html#K2" title="MATRIX_2:func.2">--&gt;</a> <font color="Maroon" title="c3">a9</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c1">n</font> <a href="matrix_2.html#K2" title="MATRIX_2:func.2">--&gt;</a> <font color="Maroon" title="c4">b9</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c1">n</font> <a href="matrix_2.html#K2" title="MATRIX_2:func.2">--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a9</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">b9</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font></span><br/><a NAME="E2:4_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c1">n</font> <a href="matrix_2.html#K2" title="MATRIX_2:func.2">--&gt;</a> <font color="Maroon" title="c3">a9</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:4"><span class="lab"><font color="Green" title="E2">A1</font></span></a>, <a class="ref" href="matrix_2.html#T1" target="_self" title="MATRIX_2:th.1">Th1</a></span>;<br/><a NAME="E3:4_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c1">n</font> <a href="matrix_2.html#K2" title="MATRIX_2:func.2">--&gt;</a> <font color="Maroon" title="c3">a9</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c1">n</font> <a href="matrix_2.html#K2" title="MATRIX_2:func.2">--&gt;</a> <font color="Maroon" title="c4">b9</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a9</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">b9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:4"><span class="lab"><font color="Green" title="E3">A2</font></span></a>, <a class="txt" href="#E1:4"><span class="lab"><font color="Green" title="E2">A1</font></span></a>, <a class="txt" href="#E1:4_1"><span class="lab"><font color="Green" title="E4">A3</font></span></a>, <a class="ref" href="matrix_2.html#T1" target="_self" title="MATRIX_2:th.1">Th1</a></span>;<br/><span class="kw">hence </span><a NAME="E4:4_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c1">n</font> <a href="matrix_2.html#K2" title="MATRIX_2:func.2">--&gt;</a> <font color="Maroon" title="c3">a9</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c1">n</font> <a href="matrix_2.html#K2" title="MATRIX_2:func.2">--&gt;</a> <font color="Maroon" title="c4">b9</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c1">n</font> <a href="matrix_2.html#K2" title="MATRIX_2:func.2">--&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a9</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">b9</font></span>)</span></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c6">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:4_1"><span class="lab"><font color="Green" title="E4">A3</font></span></a>, <a class="ref" href="matrix_2.html#T1" target="_self" title="MATRIX_2:th.1">Th1</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="E4:4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c1">n</font> <a href="matrix_2.html#K2" title="MATRIX_2:func.2">--&gt;</a> <font color="Maroon" title="c3">a9</font></span>)</span> <a href="matrix_1.html#K14" title="MATRIX_1:func.14">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c1">n</font> <a href="matrix_2.html#K2" title="MATRIX_2:func.2">--&gt;</a> <font color="Maroon" title="c4">b9</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c1">n</font> <a href="matrix_2.html#K2" title="MATRIX_2:func.2">--&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a9</font> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <font color="Maroon" title="c4">b9</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:4"><span class="lab"><font color="Green" title="E2">A1</font></span></a>, <a class="ref" href="matrix_1.html#D14" title="MATRIX_1:def.14">MATRIX_1:def 14</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/>


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