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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Olive" title="b1">K</font><br/> for <font color="Olive" title="b3">P</font>, <font color="Olive" title="b4">Q</font> being  <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">K</font>  st <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Olive" title="b1">K</font> &amp; <font color="Olive" title="b3">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a>  &amp; ( for <font color="Olive" title="b5">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b5">i</font> &amp; <font color="Olive" title="b5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/><font color="Olive" title="b3">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Olive" title="b1">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b5">i</font> ) &amp; <font color="Olive" title="b4">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> &amp; ( for <font color="Olive" title="b5">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b5">j</font> &amp; <font color="Olive" title="b5">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/><font color="Olive" title="b4">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Olive" title="b5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Olive" title="b5">j</font></span>)</span></span>)</span> ) &amp; ( for <font color="Olive" title="b5">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b5">i</font> &amp; <font color="Olive" title="b5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/><font color="Olive" title="b4">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Olive" title="b1">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b5">i</font> ) holds <br/>( <font color="Olive" title="b3">P</font> is  <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a>  &amp; <font color="Olive" title="b4">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">P</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a>  )</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">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">K</font><br/> for <font color="Olive" title="b2">P</font>, <font color="Olive" title="b3">Q</font> being  <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font>  st <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font> &amp; <font color="Olive" title="b2">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a>  &amp; ( for <font color="Olive" title="b4">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">i</font> &amp; <font color="Olive" title="b4">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/><font color="Olive" title="b2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b4">i</font> ) &amp; <font color="Olive" title="b3">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp; ( for <font color="Olive" title="b4">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">j</font> &amp; <font color="Olive" title="b4">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/><font color="Olive" title="b3">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Olive" title="b4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Olive" title="b4">j</font></span>)</span></span>)</span> ) &amp; ( for <font color="Olive" title="b4">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">i</font> &amp; <font color="Olive" title="b4">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/><font color="Olive" title="b3">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b4">i</font> ) holds <br/>( <font color="Olive" title="b2">P</font> is  <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a>  &amp; <font color="Olive" title="b3">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">P</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">a</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"> for <font color="Olive" title="b1">P</font>, <font color="Olive" title="b2">Q</font> being  <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c2">K</font>  st <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font> &amp; <font color="Olive" title="b1">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a>  &amp; ( for <font color="Olive" title="b3">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/><font color="Olive" title="b1">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b3">i</font> ) &amp; <font color="Olive" title="b2">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> &amp; ( for <font color="Olive" title="b3">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">j</font> &amp; <font color="Olive" title="b3">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/><font color="Olive" title="b2">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Olive" title="b3">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Olive" title="b3">j</font></span>)</span></span>)</span> ) &amp; ( for <font color="Olive" title="b3">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/><font color="Olive" title="b2">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b3">i</font> ) holds <br/>( <font color="Olive" title="b1">P</font> is  <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a>  &amp; <font color="Olive" title="b2">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">P</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">P</font>, <font color="Maroon" title="c5">Q</font> be   <a href="matrix_1.html#NM5" title="MATRIX_1:NM.5">Matrix</a> of <font color="Maroon" title="c1">n</font>,<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"> ( <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font> &amp; <font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a>  &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/><font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">i</font> ) &amp; <font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> &amp; ( for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Olive" title="b1">j</font></span>)</span></span>)</span> ) &amp; ( for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/><font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">i</font> ) implies ( <font color="Maroon" title="c4">P</font> is  <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a>  &amp; <font color="Maroon" title="c5">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a>  ) )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:43"/><span class="lab"><font color="Green" title="E34">A1</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">and </span><br/><a NAME="E2:43"/><span class="lab"><font color="Green" title="E35">A2</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
 <span class="kw">and </span><br/><a NAME="E3:43"/><span class="lab"><font color="Green" title="E36">A3</font></span>: 
<font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> 
 <span class="kw">and </span><br/><a NAME="E4:43"/><span class="lab"><font color="Green" title="E37">A4</font></span>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/><font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">i</font>
 <span class="kw">and </span><br/><a NAME="E5:43"/><span class="lab"><font color="Green" title="E38">A5</font></span>: 
<font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font>
 <span class="kw">and </span><br/><a NAME="E6:43"/><span class="lab"><font color="Green" title="E39">A6</font></span>: 
for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Olive" title="b1">j</font></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E7:43"/><span class="lab"><font color="Green" title="E40">A7</font></span>: 
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/><font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Olive" title="b1">i</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">P</font> is  <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a>  &amp; <font color="Maroon" title="c5">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a>  )</span><br/>

<a NAME="E8:43"/><span class="lab"><font color="Green" title="E41">A8</font></span>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E9:43"/><span class="lab"><font color="Green" title="E42">A9</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">Th23</a></span>;<br/>
<a NAME="E10:43"/><span class="lab"><font color="Green" title="E43">A10</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></span>;<br/>
<span class="lab"><font color="Green" title="E44">A11</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixr1.html#T16" title="MATRIXR1:th.16">MATRIXR1:16</a></span>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">Th23</a></span>
;<br/>
<a NAME="E12:43"/><span class="lab"><font color="Green" title="E45">A12</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> <font color="Maroon" title="c1">n</font> holds <br/><span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c4">P</font>,1</span>)</span> <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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> <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="43_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">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="c6">k</font> &amp; <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> implies <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c4">P</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> )</span><br/>

<a NAME="E1:43_2"/><span class="lab"><font color="Green" title="E46">A13</font></span>: 
<font color="Maroon" title="c6">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/>
<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:43_2"/><span class="lab"><font color="Green" title="E47">A14</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font>
 <span class="kw">and </span><br/><a NAME="E3:43_2"/><span class="lab"><font color="Green" title="E48">A15</font></span>: 
<font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</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"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c4">P</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font></span><br/>

<a NAME="E4:43_2"/><span class="lab"><font color="Green" title="E49">A16</font></span>: 
<font color="Maroon" title="c6">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> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_2"><span class="lab"><font color="Green" title="E47">A14</font></span></a>, <a class="txt" href="#E3:43_2"><span class="lab"><font color="Green" title="E48">A15</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E5:43_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A17</font></span>: 
<font color="Maroon" title="c6">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:43"><span class="lab"><font color="Green" title="E43">A10</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E51">A18</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:43_2_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">k</font>,1</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> <font color="Maroon" title="c4">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:43"><span class="lab"><font color="Green" title="E41">A8</font></span></a>, <a class="txt" href="#E2:43_2"><span class="lab"><font color="Green" title="E47">A14</font></span></a>, <a class="txt" href="#E3:43_2"><span class="lab"><font color="Green" title="E48">A15</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E2:43_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A19</font></span>: 
 ex <font color="Olive" title="b1">p</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font> st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> &amp; <font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">k</font>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</a></span>;<br/><span class="kw">assume </span><a NAME="E3:43_2_1"/><span class="lab"><font color="Green" title="E53">A20</font></span>: 
<font color="Maroon" title="c6">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <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"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c4">P</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span><br/><a NAME="E4:43_2_1"/><span class="kw">then </span>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_2"><span class="lab"><font color="Green" title="E47">A14</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">then </span><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">k</font>,1 = 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43"><span class="lab"><font color="Green" title="E37">A4</font></span></a>, <a class="txt" href="#E1:43_2"><span class="lab"><font color="Green" title="E46">A13</font></span></a>, <a class="txt" href="#E3:43_2"><span class="lab"><font color="Green" title="E48">A15</font></span></a>, <a class="txt" href="#E2:43_2_1"><span class="lab"><font color="Green" title="E52">A19</font></span></a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:43"><span class="lab"><font color="Green" title="E41">A8</font></span></a>, <a class="txt" href="#E3:43_2_1"><span class="lab"><font color="Green" title="E53">A20</font></span></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">Th25</a></span>
;<br/><span class="kw">hence </span><a NAME="E6:43_2_1"/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c4">P</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43_2"><span class="lab"><font color="Green" title="E50">A17</font></span></a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</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>
<a NAME="E7:43_2"/><span class="lab"><font color="Green" title="E52">A21</font></span>: 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:43"><span class="lab"><font color="Green" title="E42">A9</font></span></a>, <a class="txt" href="#E2:43_2"><span class="lab"><font color="Green" title="E47">A14</font></span></a>, <a class="txt" href="#E3:43_2"><span class="lab"><font color="Green" title="E48">A15</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E53">A22</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:43_2_2"/><span class="lab"><font color="Green" title="E54">A23</font></span>: 
<font color="Maroon" title="c6">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"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span><br/><a NAME="E2:43_2_2"/>
<font color="Maroon" title="c6">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:43"><span class="lab"><font color="Green" title="E44">A11</font></span></a>, <a class="txt" href="#E4:43_2"><span class="lab"><font color="Green" title="E49">A16</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/><span class="kw">hence </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:43_2"><span class="lab"><font color="Green" title="E52">A21</font></span></a>, <a class="ref" href="fvsum_1.html#T62" title="FVSUM_1:th.62">FVSUM_1:62</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_2"><span class="lab"><font color="Green" title="E48">A15</font></span></a>, <a class="txt" href="#E7:43_2"><span class="lab"><font color="Green" title="E52">A21</font></span></a>, <a class="txt" href="#E1:43_2_2"><span class="lab"><font color="Green" title="E54">A23</font></span></a>, <a class="ref" href="matrix14.html#T24" target="_self" title="MATRIX14:th.24">Th24</a></span>
<br/>.= 
<font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D16" title="VECTSP_1:def.16">VECTSP_1:def 16</a></span>
;<br/> <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>
<a NAME="E9:43_2"/><span class="lab"><font color="Green" title="E54">A24</font></span>: 
<font color="Maroon" title="c6">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:43"><span class="lab"><font color="Green" title="E44">A11</font></span></a>, <a class="txt" href="#E2:43_2"><span class="lab"><font color="Green" title="E47">A14</font></span></a>, <a class="txt" href="#E3:43_2"><span class="lab"><font color="Green" title="E48">A15</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E55">A25</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:43_2_3"/><span class="lab"><font color="Green" title="E56">A26</font></span>: 
<font color="Maroon" title="c6">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <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="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span><br/><span class="kw">thus </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:43_2"><span class="lab"><font color="Green" title="E54">A24</font></span></a>, <a class="txt" href="#E7:43_2"><span class="lab"><font color="Green" title="E52">A21</font></span></a>, <a class="ref" href="fvsum_1.html#T62" title="FVSUM_1:th.62">FVSUM_1:62</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_2"><span class="lab"><font color="Green" title="E47">A14</font></span></a>, <a class="txt" href="#E3:43_2"><span class="lab"><font color="Green" title="E48">A15</font></span></a>, <a class="txt" href="#E7:43_2"><span class="lab"><font color="Green" title="E52">A21</font></span></a>, <a class="txt" href="#E1:43_2_3"><span class="lab"><font color="Green" title="E56">A26</font></span></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">Th25</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</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>
<a NAME="E11:43_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_2"><span class="lab"><font color="Green" title="E47">A14</font></span></a>, <a class="txt" href="#E3:43_2"><span class="lab"><font color="Green" title="E48">A15</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E12:43_2"/><span class="kw">then </span>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:43"><span class="lab"><font color="Green" title="E43">A10</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:43_2"/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c4">P</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E36">A3</font></span></a>, <a class="txt" href="#E10:43_2"><span class="lab"><font color="Green" title="E55">A25</font></span></a>, <a class="txt" href="#E8:43_2"><span class="lab"><font color="Green" title="E53">A22</font></span></a>, <a class="txt" href="#E6:43_2"><span class="lab"><font color="Green" title="E51">A18</font></span></a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</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>
<a NAME="E13:43"/><span class="lab"><font color="Green" title="E46">A27</font></span>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E14:43"/><span class="lab"><font color="Green" title="E47">A28</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></span>;<br/>
<a NAME="E15:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A29</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">Q</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:43"><span class="lab"><font color="Green" title="E46">A27</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E16:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A30</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">Q</font>
 
<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="E17:43"/><span class="lab"><font color="Green" title="E50">A31</font></span>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></span>;<br/>
<a NAME="E18:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A32</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>;<br/>
<a NAME="E19:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A33</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:43"><span class="lab"><font color="Green" title="E47">A28</font></span></a>, <a class="txt" href="#E15:43"><span class="lab"><font color="Green" title="E48">A29</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E20:43"/><span class="lab"><font color="Green" title="E53">A34</font></span>: 
for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Olive" title="b1">j</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">j</font> &amp; <font color="Maroon" title="c6">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> implies <font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:43_3"/>
( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">j</font> &amp; <font color="Maroon" title="c6">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</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">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span></span><br/>

<a NAME="E2:43_3"/><span class="kw">then </span>
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:43"><span class="lab"><font color="Green" title="E39">A6</font></span></a></span>;<br/>
<a NAME="E3:43_3"/><span class="kw">then </span>
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T41" title="VECTSP_1:th.41">VECTSP_1:41</a></span>;<br/>
<a NAME="E4:43_3"/><span class="kw">then </span>
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T41" title="VECTSP_1:th.41">VECTSP_1:41</a></span>;<br/>
<a NAME="E5:43_3"/><span class="kw">then </span>
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></span>;<br/>
<a NAME="E6:43_3"/><span class="kw">then </span>
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T42" title="VECTSP_1:th.42">VECTSP_1:42</a></span>;<br/>
<a NAME="E7:43_3"/><span class="kw">then </span>
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="ref" href="vectsp_1.html#D22" title="VECTSP_1:def.22">VECTSP_1:def 22</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:43_3"/>
<font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D16" title="VECTSP_1:def.16">VECTSP_1:def 16</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>
<a NAME="E21:43"/><span class="lab"><font color="Green" title="E54">A35</font></span>: 
for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">j</font> &amp; <font color="Maroon" title="c6">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:43_4"/><span class="lab"><font color="Green" title="E55">A36</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">j</font>
 <span class="kw">and </span><br/><a NAME="E2:43_4"/><span class="lab"><font color="Green" title="E56">A37</font></span>: 
<font color="Maroon" title="c6">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</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="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span><br/>

<span class="lab"><font color="Green" title="E57">A38</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix14.html#T6" target="_self" title="MATRIX14:th.6">Th6</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixr1.html#T16" title="MATRIXR1:th.16">MATRIXR1:16</a></span>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">Th23</a></span>
;<br/>
<a NAME="E4:43_4"/><span class="lab"><font color="Green" title="E58">A39</font></span>: 
<font color="Maroon" title="c6">j</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> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_4"><span class="lab"><font color="Green" title="E55">A36</font></span></a>, <a class="txt" href="#E2:43_4"><span class="lab"><font color="Green" title="E56">A37</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E5:43_4"/><span class="lab"><font color="Green" title="E59">A40</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_4"><span class="lab"><font color="Green" title="E55">A36</font></span></a>, <a class="txt" href="#E2:43_4"><span class="lab"><font color="Green" title="E56">A37</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E6:43_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A41</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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">Q</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:43"><span class="lab"><font color="Green" title="E50">A31</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E7:43_4"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:43"><span class="lab"><font color="Green" title="E50">A31</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>;<br/>
<a NAME="E8:43_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A42</font></span>: 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_4"><span class="lab"><font color="Green" title="E55">A36</font></span></a>, <a class="txt" href="#E2:43_4"><span class="lab"><font color="Green" title="E56">A37</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/>
<span class="lab"><font color="Green" title="E62">A43</font></span>: <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:43"><span class="lab"><font color="Green" title="E51">A32</font></span></a>, <a class="txt" href="#E5:43_4"><span class="lab"><font color="Green" title="E59">A40</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>
<br/>.= 
<font color="Maroon" title="c3">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43"><span class="lab"><font color="Green" title="E38">A5</font></span></a>, <a class="txt" href="#E6:43_4"><span class="lab"><font color="Green" title="E60">A41</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>
;<br/>
<span class="lab"><font color="Green" title="E63">A44</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">j</font></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">P</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></span>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></span>
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">p</font> =  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">j</font>, <font color="Maroon" title="c8">q</font> = <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font> ;<br/>
<a NAME="E12:43_4"/><span class="lab"><font color="Green" title="E64">A45</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">Th23</a></span>;<br/>
<span class="lab"><font color="Green" title="E65">A46</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixr1.html#T16" title="MATRIXR1:th.16">MATRIXR1:16</a></span>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">Th23</a></span>
;<br/>
<a NAME="E14:43_4"/><span class="lab"><font color="Green" title="E66">A47</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> <font color="Maroon" title="c1">n</font> holds <br/><font color="Maroon" title="c7">p</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="c8">q</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="43_4_5"><span class="kw">proof </span></a><div class="add">

<span class="lab"><font color="Green" title="E67">A48</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixr1.html#T16" title="MATRIXR1:th.16">MATRIXR1:16</a></span>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">Th23</a></span>
;<br/>
<span class="kw">let </span><font color="Maroon" title="c9">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="c9">k</font> &amp; <font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> implies <font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:43_4_5"/><span class="lab"><font color="Green" title="E68">A49</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">k</font>
 <span class="kw">and </span><br/><a NAME="E3:43_4_5"/><span class="lab"><font color="Green" title="E69">A50</font></span>: 
<font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</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="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font></span><br/>

<a NAME="E4:43_4_5"/><span class="lab"><font color="Green" title="E70">A51</font></span>: 
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:43_4"><span class="lab"><font color="Green" title="E65">A46</font></span></a>, <a class="txt" href="#E2:43_4_5"><span class="lab"><font color="Green" title="E68">A49</font></span></a>, <a class="txt" href="#E3:43_4_5"><span class="lab"><font color="Green" title="E69">A50</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E5:43_4_5"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">Th23</a></span>;<br/>
<a NAME="E6:43_4_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A52</font></span>: 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_4_5"><span class="lab"><font color="Green" title="E68">A49</font></span></a>, <a class="txt" href="#E3:43_4_5"><span class="lab"><font color="Green" title="E69">A50</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/>
<a NAME="E7:43_4_5"/>
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:43"><span class="lab"><font color="Green" title="E43">A10</font></span></a>, <a class="txt" href="#E2:43_4_5"><span class="lab"><font color="Green" title="E68">A49</font></span></a>, <a class="txt" href="#E3:43_4_5"><span class="lab"><font color="Green" title="E69">A50</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E8:43_4_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A53</font></span>: 
<font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c6">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></span>;<br/>
<span class="lab"><font color="Green" title="E73">A54</font></span>: <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix14.html#T6" target="_self" title="MATRIX14:th.6">Th6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_4_5"><span class="lab"><font color="Green" title="E70">A51</font></span></a>, <a class="txt" href="#E6:43_4_5"><span class="lab"><font color="Green" title="E71">A52</font></span></a>, <a class="ref" href="fvsum_1.html#T62" title="FVSUM_1:th.62">FVSUM_1:62</a></span>
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_4_5_3"><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="E1:43_4_5_3"/>
( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k</font> or 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">k</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_4_5"><span class="lab"><font color="Green" title="E68">A49</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_4_5_3_1"><span class="kw">suppose </span></a><a NAME="E1:43_4_5_3_1"/><span class="lab"><font color="Green" title="E74">A55</font></span>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k</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="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font></span><br/><div class="add"><a NAME="E2:43_4_5_3_1"/><span class="kw">then </span>
1 <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> <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_4_5"><span class="lab"><font color="Green" title="E69">A50</font></span></a>, <a class="txt" href="#E1:43_4_5"><span class="lab"><font color="Green" title="E67">A48</font></span></a>, <a class="ref" href="matrix14.html#T6" target="_self" title="MATRIX14:th.6">Th6</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A56</font></span>: <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:43_4_5"><span class="lab"><font color="Green" title="E73">A54</font></span></a>, <a class="txt" href="#E1:43_4_5_3_1"><span class="lab"><font color="Green" title="E74">A55</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:43"><span class="lab"><font color="Green" title="E46">A27</font></span></a>, <a class="txt" href="#E6:43_4_5"><span class="lab"><font color="Green" title="E71">A52</font></span></a>, <a class="txt" href="#E1:43_4_5_3_1"><span class="lab"><font color="Green" title="E74">A55</font></span></a>, <a class="ref" href="matrix14.html#T24" target="_self" title="MATRIX14:th.24">Th24</a></span>
<br/>.= 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D16" title="VECTSP_1:def.16">VECTSP_1:def 16</a></span>
;<br/><a NAME="E4:43_4_5_3_1"/><span class="lab"><font color="Green" title="E76">A57</font></span>: 
<font color="Maroon" title="c7">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>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:43"><span class="lab"><font color="Green" title="E53">A34</font></span></a>, <a class="txt" href="#E1:43_4"><span class="lab"><font color="Green" title="E55">A36</font></span></a>, <a class="txt" href="#E2:43_4"><span class="lab"><font color="Green" title="E56">A37</font></span></a>, <a class="txt" href="#E8:43_4_5"><span class="lab"><font color="Green" title="E72">A53</font></span></a>, <a class="txt" href="#E1:43_4_5_3_1"><span class="lab"><font color="Green" title="E74">A55</font></span></a></span>;<br/><span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:43_4"><span class="lab"><font color="Green" title="E64">A45</font></span></a>, <a class="txt" href="#E3:43_4_5"><span class="lab"><font color="Green" title="E69">A50</font></span></a>, <a class="txt" href="#E1:43_4_5_3_1"><span class="lab"><font color="Green" title="E74">A55</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_4"><span class="lab"><font color="Green" title="E55">A36</font></span></a>, <a class="txt" href="#E3:43_4_5"><span class="lab"><font color="Green" title="E69">A50</font></span></a>, <a class="txt" href="#E1:43_4_5_3_1"><span class="lab"><font color="Green" title="E74">A55</font></span></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">Th25</a></span>
;<br/><a NAME="E6:43_4_5_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_4"><span class="lab"><font color="Green" title="E57">A38</font></span></a>, <a class="txt" href="#E12:43_4"><span class="lab"><font color="Green" title="E64">A45</font></span></a>, <a class="txt" href="#E3:43_4_5"><span class="lab"><font color="Green" title="E69">A50</font></span></a>, <a class="txt" href="#E1:43_4_5_3_1"><span class="lab"><font color="Green" title="E74">A55</font></span></a>, <a class="txt" href="#E3:43_4_5_3_1"><span class="lab"><font color="Green" title="E75">A56</font></span></a>, <a class="ref" href="matrix14.html#T5" target="_self" title="MATRIX14:th.5">Th5</a></span>;<br/><span class="kw">hence </span><a NAME="E7:43_4_5_3_1"/>
<font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_4_5_3_1"><span class="lab"><font color="Green" title="E74">A55</font></span></a>, <a class="txt" href="#E4:43_4_5_3_1"><span class="lab"><font color="Green" title="E76">A57</font></span></a>, <a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</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="43_4_5_3_2"><span class="kw">suppose </span></a><a NAME="E1:43_4_5_3_2"/><span class="lab"><font color="Green" title="E74">A58</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">k</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="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font></span><br/><div class="add"><a NAME="E2:43_4_5_3_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">k</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> <font color="Maroon" title="c4">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_4"><span class="lab"><font color="Green" title="E55">A36</font></span></a>, <a class="txt" href="#E2:43_4"><span class="lab"><font color="Green" title="E56">A37</font></span></a>, <a class="txt" href="#E2:43_4_5"><span class="lab"><font color="Green" title="E68">A49</font></span></a>, <a class="txt" href="#E3:43_4_5"><span class="lab"><font color="Green" title="E69">A50</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E3:43_4_5_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A59</font></span>: 
 ex <font color="Olive" title="b1">p2</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font> st <br/>( <font color="Olive" title="b1">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> &amp; <font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</a></span>;<br/><a NAME="E4:43_4_5_3_2"/>
<font color="Maroon" title="c9">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="E5:43_4_5_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A60</font></span>: 
<font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c9">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43"><span class="lab"><font color="Green" title="E37">A4</font></span></a>, <a class="txt" href="#E3:43_4_5"><span class="lab"><font color="Green" title="E69">A50</font></span></a>, <a class="txt" href="#E8:43_4_5"><span class="lab"><font color="Green" title="E72">A53</font></span></a>, <a class="txt" href="#E1:43_4_5_3_2"><span class="lab"><font color="Green" title="E74">A58</font></span></a>, <a class="txt" href="#E3:43_4_5_3_2"><span class="lab"><font color="Green" title="E75">A59</font></span></a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_4_5_3_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_4_5_3_2_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="E1:43_4_5_3_2_1_1"/>
( <font color="Maroon" title="c9">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">j</font> or <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_4_5_3_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:43_4_5_3_2_1_1_1"/><span class="lab"><font color="Green" title="E77">A61</font></span>: 
<font color="Maroon" title="c9">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">j</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="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font></span><br/><div class="add"><a NAME="E2:43_4_5_3_2_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">Th23</a></span>;<br/><a NAME="E3:43_4_5_3_2_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_4_5"><span class="lab"><font color="Green" title="E68">A49</font></span></a>, <a class="txt" href="#E3:43_4_5"><span class="lab"><font color="Green" title="E69">A50</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A62</font></span>: <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> = 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_4_5"><span class="lab"><font color="Green" title="E69">A50</font></span></a>, <a class="txt" href="#E9:43_4_5"><span class="lab"><font color="Green" title="E73">A54</font></span></a>, <a class="txt" href="#E1:43_4_5_3_2"><span class="lab"><font color="Green" title="E74">A58</font></span></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">Th25</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></span>
;<br/><a NAME="E5:43_4_5_3_2_1_1_1"/><span class="lab"><font color="Green" title="E79">A63</font></span>: 
<span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_4"><span class="lab"><font color="Green" title="E57">A38</font></span></a>, <a class="txt" href="#E2:43_4_5"><span class="lab"><font color="Green" title="E68">A49</font></span></a>, <a class="txt" href="#E3:43_4_5"><span class="lab"><font color="Green" title="E69">A50</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/><a NAME="E6:43_4_5_3_2_1_1_1"/><span class="lab"><font color="Green" title="E80">A64</font></span>: 
<font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_4"><span class="lab"><font color="Green" title="E55">A36</font></span></a>, <a class="txt" href="#E2:43_4"><span class="lab"><font color="Green" title="E56">A37</font></span></a>, <a class="txt" href="#E5:43_4_5_3_2"><span class="lab"><font color="Green" title="E76">A60</font></span></a>, <a class="txt" href="#E1:43_4_5_3_2_1_1_1"><span class="lab"><font color="Green" title="E77">A61</font></span></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">Th25</a></span>;<br/><span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font> = 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:43_4"><span class="lab"><font color="Green" title="E64">A45</font></span></a>, <a class="txt" href="#E2:43_4_5"><span class="lab"><font color="Green" title="E68">A49</font></span></a>, <a class="txt" href="#E3:43_4_5"><span class="lab"><font color="Green" title="E69">A50</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_4_5"><span class="lab"><font color="Green" title="E68">A49</font></span></a>, <a class="txt" href="#E3:43_4_5"><span class="lab"><font color="Green" title="E69">A50</font></span></a>, <a class="txt" href="#E1:43_4_5_3_2_1_1_1"><span class="lab"><font color="Green" title="E77">A61</font></span></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">Th25</a></span>
;<br/><a NAME="E8:43_4_5_3_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_4"><span class="lab"><font color="Green" title="E57">A38</font></span></a>, <a class="txt" href="#E12:43_4"><span class="lab"><font color="Green" title="E64">A45</font></span></a>, <a class="txt" href="#E2:43_4_5"><span class="lab"><font color="Green" title="E68">A49</font></span></a>, <a class="txt" href="#E3:43_4_5"><span class="lab"><font color="Green" title="E69">A50</font></span></a>, <a class="txt" href="#E5:43_4_5_3_2_1_1_1"><span class="lab"><font color="Green" title="E79">A63</font></span></a>, <a class="txt" href="#E4:43_4_5_3_2_1_1_1"><span class="lab"><font color="Green" title="E78">A62</font></span></a>, <a class="ref" href="matrix14.html#T5" target="_self" title="MATRIX14:th.5">Th5</a></span>;<br/><span class="kw">hence </span><a NAME="E9:43_4_5_3_2_1_1_1"/>
<font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:43_4_5_3_2_1_1_1"><span class="lab"><font color="Green" title="E80">A64</font></span></a>, <a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</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="43_4_5_3_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:43_4_5_3_2_1_1_2"/><span class="lab"><font color="Green" title="E77">A65</font></span>: 
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">j</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="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font></span><br/><div class="add"><a NAME="E2:43_4_5_3_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A66</font></span>: 
<font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_4_5"><span class="lab"><font color="Green" title="E68">A49</font></span></a>, <a class="txt" href="#E3:43_4_5"><span class="lab"><font color="Green" title="E69">A50</font></span></a>, <a class="txt" href="#E5:43_4_5_3_2"><span class="lab"><font color="Green" title="E76">A60</font></span></a>, <a class="ref" href="matrix14.html#T24" target="_self" title="MATRIX14:th.24">Th24</a></span>;<br/><a NAME="E3:43_4_5_3_2_1_1_2"/><span class="lab"><font color="Green" title="E79">A67</font></span>: 
<span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c9">k</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c9">k</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_4"><span class="lab"><font color="Green" title="E57">A38</font></span></a>, <a class="txt" href="#E2:43_4_5"><span class="lab"><font color="Green" title="E68">A49</font></span></a>, <a class="txt" href="#E3:43_4_5"><span class="lab"><font color="Green" title="E69">A50</font></span></a>, <a class="txt" href="#E1:43_4_5_3_2_1_1_2"><span class="lab"><font color="Green" title="E77">A65</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/><a NAME="E4:43_4_5_3_2_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">Th23</a></span>;<br/><a NAME="E5:43_4_5_3_2_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_4_5"><span class="lab"><font color="Green" title="E68">A49</font></span></a>, <a class="txt" href="#E3:43_4_5"><span class="lab"><font color="Green" title="E69">A50</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A68</font></span>: <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c9">k</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> = 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c9">k</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_4_5"><span class="lab"><font color="Green" title="E69">A50</font></span></a>, <a class="txt" href="#E9:43_4_5"><span class="lab"><font color="Green" title="E73">A54</font></span></a>, <a class="txt" href="#E1:43_4_5_3_2"><span class="lab"><font color="Green" title="E74">A58</font></span></a>, <a class="txt" href="#E1:43_4_5_3_2_1_1_2"><span class="lab"><font color="Green" title="E77">A65</font></span></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">Th25</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></span>
;<br/><span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c9">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font> = 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c9">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:43_4"><span class="lab"><font color="Green" title="E64">A45</font></span></a>, <a class="txt" href="#E2:43_4_5"><span class="lab"><font color="Green" title="E68">A49</font></span></a>, <a class="txt" href="#E3:43_4_5"><span class="lab"><font color="Green" title="E69">A50</font></span></a>, <a class="txt" href="#E1:43_4_5_3_2_1_1_2"><span class="lab"><font color="Green" title="E77">A65</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>
<br/>.= 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_4"><span class="lab"><font color="Green" title="E55">A36</font></span></a>, <a class="txt" href="#E2:43_4"><span class="lab"><font color="Green" title="E56">A37</font></span></a>, <a class="txt" href="#E1:43_4_5_3_2_1_1_2"><span class="lab"><font color="Green" title="E77">A65</font></span></a>, <a class="ref" href="matrix14.html#T24" target="_self" title="MATRIX14:th.24">Th24</a></span>
;<br/><a NAME="E8:43_4_5_3_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_4"><span class="lab"><font color="Green" title="E57">A38</font></span></a>, <a class="txt" href="#E12:43_4"><span class="lab"><font color="Green" title="E64">A45</font></span></a>, <a class="txt" href="#E2:43_4_5"><span class="lab"><font color="Green" title="E68">A49</font></span></a>, <a class="txt" href="#E3:43_4_5"><span class="lab"><font color="Green" title="E69">A50</font></span></a>, <a class="txt" href="#E1:43_4_5_3_2_1_1_2"><span class="lab"><font color="Green" title="E77">A65</font></span></a>, <a class="txt" href="#E3:43_4_5_3_2_1_1_2"><span class="lab"><font color="Green" title="E79">A67</font></span></a>, <a class="txt" href="#E6:43_4_5_3_2_1_1_2"><span class="lab"><font color="Green" title="E80">A68</font></span></a>, <a class="ref" href="matrix14.html#T5" target="_self" title="MATRIX14:th.5">Th5</a></span>;<br/><span class="kw">hence </span><a NAME="E9:43_4_5_3_2_1_1_2"/>
<font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_4_5_3_2_1_1_2"><span class="lab"><font color="Green" title="E78">A66</font></span></a>, <a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</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><span class="kw">hence </span><a NAME="E7:43_4_5_3_2"/>
<font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font>
 ; <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 NAME="E15:43_4"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p4">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_4"><span class="lab"><font color="Green" title="E57">A38</font></span></a>, <a class="txt" href="#E12:43_4"><span class="lab"><font color="Green" title="E64">A45</font></span></a>, <a class="ref" href="matrix14.html#T2" target="_self" title="MATRIX14:th.2">Th2</a></span>;<br/>
<a NAME="E16:43_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A69</font></span>: 
 <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:43_4"><span class="lab"><font color="Green" title="E63">A44</font></span></a>, <a class="txt" href="#E14:43_4"><span class="lab"><font color="Green" title="E66">A47</font></span></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></span>;<br/>
<a NAME="E17:43_4"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,<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="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:43"><span class="lab"><font color="Green" title="E46">A27</font></span></a>, <a class="txt" href="#E1:43_4"><span class="lab"><font color="Green" title="E55">A36</font></span></a>, <a class="txt" href="#E2:43_4"><span class="lab"><font color="Green" title="E56">A37</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font> = 
<span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:43"><span class="lab"><font color="Green" title="E43">A10</font></span></a>, <a class="txt" href="#E17:43"><span class="lab"><font color="Green" title="E50">A31</font></span></a>, <a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a></span>
<br/>.= 
<span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p4">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:43"><span class="lab"><font color="Green" title="E51">A32</font></span></a>, <a class="txt" href="#E3:43_4"><span class="lab"><font color="Green" title="E57">A38</font></span></a>, <a class="txt" href="#E12:43_4"><span class="lab"><font color="Green" title="E64">A45</font></span></a>, <a class="txt" href="#E16:43_4"><span class="lab"><font color="Green" title="E67">A69</font></span></a>, <a class="ref" href="matrix14.html#T11" target="_self" title="MATRIX14:th.11">Th11</a></span>
<br/>.= 
<span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix14.html#T6" target="_self" title="MATRIX14:th.6">Th6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span>,<span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:43"><span class="lab"><font color="Green" title="E51">A32</font></span></a>, <a class="txt" href="#E9:43"><span class="lab"><font color="Green" title="E42">A9</font></span></a>, <a class="ref" href="matrix14.html#T10" target="_self" title="MATRIX14:th.10">Th10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">a</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:43"><span class="lab"><font color="Green" title="E51">A32</font></span></a>, <a class="txt" href="#E13:43"><span class="lab"><font color="Green" title="E46">A27</font></span></a>, <a class="txt" href="#E9:43_4"><span class="lab"><font color="Green" title="E62">A43</font></span></a>, <a class="ref" href="matrix14.html#T35" target="_self" title="MATRIX14:th.35">Th35</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T41" title="VECTSP_1:th.41">VECTSP_1:41</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c3">a</font></span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="ref" href="vectsp_1.html#D22" title="VECTSP_1:def.22">VECTSP_1:def 22</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span></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"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:43"><span class="lab"><font color="Green" title="E51">A32</font></span></a>, <a class="txt" href="#E1:43_4"><span class="lab"><font color="Green" title="E55">A36</font></span></a>, <a class="txt" href="#E2:43_4"><span class="lab"><font color="Green" title="E56">A37</font></span></a>, <a class="ref" href="matrix14.html#T35" target="_self" title="MATRIX14:th.35">Th35</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></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"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D16" title="VECTSP_1:def.16">VECTSP_1:def 16</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:43"><span class="lab"><font color="Green" title="E50">A31</font></span></a>, <a class="txt" href="#E4:43_4"><span class="lab"><font color="Green" title="E58">A39</font></span></a>, <a class="txt" href="#E8:43_4"><span class="lab"><font color="Green" title="E61">A42</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T16" title="RLVECT_1:th.16">RLVECT_1:16</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E19:43_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E22:43"/><span class="lab"><font color="Green" title="E55">A70</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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">Q</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:43"><span class="lab"><font color="Green" title="E50">A31</font></span></a>, <a class="txt" href="#E13:43"><span class="lab"><font color="Green" title="E46">A27</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E23:43"/><span class="lab"><font color="Green" title="E56">A71</font></span>: 
for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">j</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">i</font>, <font color="Maroon" title="c7">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">i</font> &amp; <font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:43_5"/><span class="lab"><font color="Green" title="E57">A72</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">i</font>
 <span class="kw">and </span><br/><a NAME="E2:43_5"/><span class="lab"><font color="Green" title="E58">A73</font></span>: 
<font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">and </span><br/><a NAME="E3:43_5"/><span class="lab"><font color="Green" title="E59">A74</font></span>: 
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</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="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font></span><br/>

<span class="lab"><font color="Green" title="E60">A75</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix14.html#T6" target="_self" title="MATRIX14:th.6">Th6</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixr1.html#T16" title="MATRIXR1:th.16">MATRIXR1:16</a></span>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">Th23</a></span>
;<br/>
<a NAME="E5:43_5"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">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> <font color="Maroon" title="c5">Q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_5"><span class="lab"><font color="Green" title="E57">A72</font></span></a>, <a class="txt" href="#E2:43_5"><span class="lab"><font color="Green" title="E58">A73</font></span></a>, <a class="txt" href="#E3:43_5"><span class="lab"><font color="Green" title="E59">A74</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">p1</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font><span class="kw"> such that </span><br/><a NAME="E7:43_5"/><span class="lab"><font color="Green" title="E61">A76</font></span>: 
<font color="Maroon" title="c8">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
 <span class="kw">and </span><br/><a NAME="E8:43_5"/><span class="lab"><font color="Green" title="E62">A77</font></span>: 
<font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</a></span>;<br/>
<a NAME="E9:43_5"/>
<font color="Maroon" title="c8">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:43"><span class="lab"><font color="Green" title="E40">A7</font></span></a>, <a class="txt" href="#E1:43_5"><span class="lab"><font color="Green" title="E57">A72</font></span></a>, <a class="txt" href="#E2:43_5"><span class="lab"><font color="Green" title="E58">A73</font></span></a>, <a class="txt" href="#E7:43_5"><span class="lab"><font color="Green" title="E61">A76</font></span></a></span>;<br/>
<a NAME="E10:43_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A78</font></span>: 
<font color="Maroon" title="c8">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_5"><span class="lab"><font color="Green" title="E57">A72</font></span></a>, <a class="txt" href="#E2:43_5"><span class="lab"><font color="Green" title="E58">A73</font></span></a>, <a class="txt" href="#E3:43_5"><span class="lab"><font color="Green" title="E59">A74</font></span></a>, <a class="ref" href="matrix14.html#T24" target="_self" title="MATRIX14:th.24">Th24</a></span>;<br/>
<a NAME="E11:43_5"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">i</font>,1</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> <font color="Maroon" title="c5">Q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:43"><span class="lab"><font color="Green" title="E41">A8</font></span></a>, <a class="txt" href="#E1:43_5"><span class="lab"><font color="Green" title="E57">A72</font></span></a>, <a class="txt" href="#E2:43_5"><span class="lab"><font color="Green" title="E58">A73</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">p2</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font><span class="kw"> such that </span><br/><a NAME="E13:43_5"/><span class="lab"><font color="Green" title="E64">A79</font></span>: 
<font color="Maroon" title="c9">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
 <span class="kw">and </span><br/><a NAME="E14:43_5"/><span class="lab"><font color="Green" title="E65">A80</font></span>: 
<font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</a></span>;<br/>
<a NAME="E15:43_5"/><span class="lab"><font color="Green" title="E66">A81</font></span>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></span>;<br/>
<a NAME="E16:43_5"/><span class="lab"><font color="Green" title="E67">A82</font></span>: 
<font color="Maroon" title="c7">j</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> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_5"><span class="lab"><font color="Green" title="E57">A72</font></span></a>, <a class="txt" href="#E2:43_5"><span class="lab"><font color="Green" title="E58">A73</font></span></a>, <a class="txt" href="#E3:43_5"><span class="lab"><font color="Green" title="E59">A74</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<span class="lab"><font color="Green" title="E68">A83</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c7">j</font></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">P</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></span>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></span>
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">p</font> =  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c7">j</font>, <font color="Maroon" title="c11">q</font> = <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font> ;<br/>
<a NAME="E19:43_5"/><span class="lab"><font color="Green" title="E69">A84</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">Th23</a></span>;<br/>
<span class="lab"><font color="Green" title="E70">A85</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixr1.html#T16" title="MATRIXR1:th.16">MATRIXR1:16</a></span>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">Th23</a></span>
;<br/>
<a NAME="E21:43_5"/><span class="lab"><font color="Green" title="E71">A86</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> <font color="Maroon" title="c1">n</font> holds <br/><font color="Maroon" title="c10">p</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="c11">q</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="43_5_4"><span class="kw">proof </span></a><div class="add">

<span class="lab"><font color="Green" title="E72">A87</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixr1.html#T16" title="MATRIXR1:th.16">MATRIXR1:16</a></span>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">Th23</a></span>
;<br/>
<span class="kw">let </span><font color="Maroon" title="c12">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="c12">k</font> &amp; <font color="Maroon" title="c12">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> implies <font color="Maroon" title="c10">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:43_5_4"/><span class="lab"><font color="Green" title="E73">A88</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">k</font>
 <span class="kw">and </span><br/><a NAME="E3:43_5_4"/><span class="lab"><font color="Green" title="E74">A89</font></span>: 
<font color="Maroon" title="c12">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</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="c10">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font></span><br/>

<a NAME="E4:43_5_4"/>
<font color="Maroon" title="c12">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> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_5_4"><span class="lab"><font color="Green" title="E73">A88</font></span></a>, <a class="txt" href="#E3:43_5_4"><span class="lab"><font color="Green" title="E74">A89</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E5:43_5_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A90</font></span>: 
<font color="Maroon" title="c12">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:43_5"><span class="lab"><font color="Green" title="E70">A85</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E6:43_5_4"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">Th23</a></span>;<br/>
<a NAME="E7:43_5_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A91</font></span>: 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_5_4"><span class="lab"><font color="Green" title="E73">A88</font></span></a>, <a class="txt" href="#E3:43_5_4"><span class="lab"><font color="Green" title="E74">A89</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/>
<a NAME="E8:43_5_4"/>
<font color="Maroon" title="c12">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:43"><span class="lab"><font color="Green" title="E43">A10</font></span></a>, <a class="txt" href="#E2:43_5_4"><span class="lab"><font color="Green" title="E73">A88</font></span></a>, <a class="txt" href="#E3:43_5_4"><span class="lab"><font color="Green" title="E74">A89</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E9:43_5_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A92</font></span>: 
<font color="Maroon" title="c10">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">k</font>,<font color="Maroon" title="c7">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></span>;<br/>
<span class="lab"><font color="Green" title="E78">A93</font></span>: <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix14.html#T6" target="_self" title="MATRIX14:th.6">Th6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43_5_4"><span class="lab"><font color="Green" title="E75">A90</font></span></a>, <a class="txt" href="#E7:43_5_4"><span class="lab"><font color="Green" title="E76">A91</font></span></a>, <a class="ref" href="fvsum_1.html#T62" title="FVSUM_1:th.62">FVSUM_1:62</a></span>
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_5_4_3"><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="E1:43_5_4_3"/>
( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">k</font> or 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">k</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_5_4"><span class="lab"><font color="Green" title="E73">A88</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_5_4_3_1"><span class="kw">suppose </span></a><a NAME="E1:43_5_4_3_1"/><span class="lab"><font color="Green" title="E79">A94</font></span>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">k</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="c10">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font></span><br/><div class="add"><a NAME="E2:43_5_4_3_1"/>
<font color="Maroon" title="c12">k</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> <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_5_4"><span class="lab"><font color="Green" title="E74">A89</font></span></a>, <a class="txt" href="#E1:43_5_4"><span class="lab"><font color="Green" title="E72">A87</font></span></a>, <a class="ref" href="matrix14.html#T6" target="_self" title="MATRIX14:th.6">Th6</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A95</font></span>: <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">k</font> = 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_5_4"><span class="lab"><font color="Green" title="E73">A88</font></span></a>, <a class="txt" href="#E10:43_5_4"><span class="lab"><font color="Green" title="E78">A93</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:43"><span class="lab"><font color="Green" title="E46">A27</font></span></a>, <a class="txt" href="#E7:43_5_4"><span class="lab"><font color="Green" title="E76">A91</font></span></a>, <a class="txt" href="#E1:43_5_4_3_1"><span class="lab"><font color="Green" title="E79">A94</font></span></a>, <a class="ref" href="matrix14.html#T24" target="_self" title="MATRIX14:th.24">Th24</a></span>
<br/>.= 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D16" title="VECTSP_1:def.16">VECTSP_1:def 16</a></span>
;<br/><span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">k</font> = 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:43_5"><span class="lab"><font color="Green" title="E69">A84</font></span></a>, <a class="txt" href="#E2:43_5_4"><span class="lab"><font color="Green" title="E73">A88</font></span></a>, <a class="txt" href="#E3:43_5_4"><span class="lab"><font color="Green" title="E74">A89</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_5"><span class="lab"><font color="Green" title="E57">A72</font></span></a>, <a class="txt" href="#E3:43_5"><span class="lab"><font color="Green" title="E59">A74</font></span></a>, <a class="txt" href="#E3:43_5_4"><span class="lab"><font color="Green" title="E74">A89</font></span></a>, <a class="txt" href="#E1:43_5_4_3_1"><span class="lab"><font color="Green" title="E79">A94</font></span></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">Th25</a></span>
;<br/><a NAME="E5:43_5_4_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A96</font></span>: 
<font color="Maroon" title="c11">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_5"><span class="lab"><font color="Green" title="E60">A75</font></span></a>, <a class="txt" href="#E19:43_5"><span class="lab"><font color="Green" title="E69">A84</font></span></a>, <a class="txt" href="#E2:43_5_4"><span class="lab"><font color="Green" title="E73">A88</font></span></a>, <a class="txt" href="#E3:43_5_4"><span class="lab"><font color="Green" title="E74">A89</font></span></a>, <a class="txt" href="#E3:43_5_4_3_1"><span class="lab"><font color="Green" title="E80">A95</font></span></a>, <a class="ref" href="matrix14.html#T5" target="_self" title="MATRIX14:th.5">Th5</a></span>;<br/><a NAME="E6:43_5_4_3_1"/>
<font color="Maroon" title="c10">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:43"><span class="lab"><font color="Green" title="E53">A34</font></span></a>, <a class="txt" href="#E1:43_5"><span class="lab"><font color="Green" title="E57">A72</font></span></a>, <a class="txt" href="#E2:43_5"><span class="lab"><font color="Green" title="E58">A73</font></span></a>, <a class="txt" href="#E3:43_5"><span class="lab"><font color="Green" title="E59">A74</font></span></a>, <a class="txt" href="#E9:43_5_4"><span class="lab"><font color="Green" title="E77">A92</font></span></a>, <a class="txt" href="#E1:43_5_4_3_1"><span class="lab"><font color="Green" title="E79">A94</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:43_5_4_3_1"/>
<font color="Maroon" title="c10">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43_5_4_3_1"><span class="lab"><font color="Green" title="E81">A96</font></span></a>, <a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</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="43_5_4_3_2"><span class="kw">suppose </span></a><a NAME="E1:43_5_4_3_2"/><span class="lab"><font color="Green" title="E79">A97</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">k</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="c10">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font></span><br/><div class="add"><a NAME="E2:43_5_4_3_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">k</font>,<font color="Maroon" title="c7">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> <font color="Maroon" title="c4">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_5"><span class="lab"><font color="Green" title="E57">A72</font></span></a>, <a class="txt" href="#E2:43_5"><span class="lab"><font color="Green" title="E58">A73</font></span></a>, <a class="txt" href="#E3:43_5"><span class="lab"><font color="Green" title="E59">A74</font></span></a>, <a class="txt" href="#E2:43_5_4"><span class="lab"><font color="Green" title="E73">A88</font></span></a>, <a class="txt" href="#E3:43_5_4"><span class="lab"><font color="Green" title="E74">A89</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E3:43_5_4_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E80">A98</font></span>: 
 ex <font color="Olive" title="b1">p2</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font> st <br/>( <font color="Olive" title="b1">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font> &amp; <font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c12">k</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</a></span>;<br/><a NAME="E4:43_5_4_3_2"/>
<font color="Maroon" title="c12">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="E5:43_5_4_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A99</font></span>: 
<font color="Maroon" title="c10">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c12">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43"><span class="lab"><font color="Green" title="E37">A4</font></span></a>, <a class="txt" href="#E3:43_5_4"><span class="lab"><font color="Green" title="E74">A89</font></span></a>, <a class="txt" href="#E9:43_5_4"><span class="lab"><font color="Green" title="E77">A92</font></span></a>, <a class="txt" href="#E1:43_5_4_3_2"><span class="lab"><font color="Green" title="E79">A97</font></span></a>, <a class="txt" href="#E3:43_5_4_3_2"><span class="lab"><font color="Green" title="E80">A98</font></span></a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_5_4_3_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_5_4_3_2_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="E1:43_5_4_3_2_1_1"/>
( <font color="Maroon" title="c12">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">j</font> or <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_5_4_3_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:43_5_4_3_2_1_1_1"/><span class="lab"><font color="Green" title="E82">A100</font></span>: 
<font color="Maroon" title="c12">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">j</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="c10">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font></span><br/><div class="add"><a NAME="E2:43_5_4_3_2_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">Th23</a></span>;<br/><a NAME="E3:43_5_4_3_2_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_5_4"><span class="lab"><font color="Green" title="E73">A88</font></span></a>, <a class="txt" href="#E3:43_5_4"><span class="lab"><font color="Green" title="E74">A89</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A101</font></span>: <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font> = 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_5_4"><span class="lab"><font color="Green" title="E74">A89</font></span></a>, <a class="txt" href="#E10:43_5_4"><span class="lab"><font color="Green" title="E78">A93</font></span></a>, <a class="txt" href="#E1:43_5_4_3_2"><span class="lab"><font color="Green" title="E79">A97</font></span></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">Th25</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></span>
;<br/><a NAME="E5:43_5_4_3_2_1_1_1"/><span class="lab"><font color="Green" title="E84">A102</font></span>: 
<span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_5"><span class="lab"><font color="Green" title="E60">A75</font></span></a>, <a class="txt" href="#E2:43_5_4"><span class="lab"><font color="Green" title="E73">A88</font></span></a>, <a class="txt" href="#E3:43_5_4"><span class="lab"><font color="Green" title="E74">A89</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/><a NAME="E6:43_5_4_3_2_1_1_1"/><span class="lab"><font color="Green" title="E85">A103</font></span>: 
<font color="Maroon" title="c10">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_5"><span class="lab"><font color="Green" title="E57">A72</font></span></a>, <a class="txt" href="#E2:43_5"><span class="lab"><font color="Green" title="E58">A73</font></span></a>, <a class="txt" href="#E3:43_5"><span class="lab"><font color="Green" title="E59">A74</font></span></a>, <a class="txt" href="#E5:43_5_4_3_2"><span class="lab"><font color="Green" title="E81">A99</font></span></a>, <a class="txt" href="#E1:43_5_4_3_2_1_1_1"><span class="lab"><font color="Green" title="E82">A100</font></span></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">Th25</a></span>;<br/><span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">k</font> = 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:43_5"><span class="lab"><font color="Green" title="E69">A84</font></span></a>, <a class="txt" href="#E2:43_5_4"><span class="lab"><font color="Green" title="E73">A88</font></span></a>, <a class="txt" href="#E3:43_5_4"><span class="lab"><font color="Green" title="E74">A89</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_5_4"><span class="lab"><font color="Green" title="E73">A88</font></span></a>, <a class="txt" href="#E3:43_5_4"><span class="lab"><font color="Green" title="E74">A89</font></span></a>, <a class="txt" href="#E1:43_5_4_3_2_1_1_1"><span class="lab"><font color="Green" title="E82">A100</font></span></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">Th25</a></span>
;<br/><a NAME="E8:43_5_4_3_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_5"><span class="lab"><font color="Green" title="E60">A75</font></span></a>, <a class="txt" href="#E19:43_5"><span class="lab"><font color="Green" title="E69">A84</font></span></a>, <a class="txt" href="#E2:43_5_4"><span class="lab"><font color="Green" title="E73">A88</font></span></a>, <a class="txt" href="#E3:43_5_4"><span class="lab"><font color="Green" title="E74">A89</font></span></a>, <a class="txt" href="#E5:43_5_4_3_2_1_1_1"><span class="lab"><font color="Green" title="E84">A102</font></span></a>, <a class="txt" href="#E4:43_5_4_3_2_1_1_1"><span class="lab"><font color="Green" title="E83">A101</font></span></a>, <a class="ref" href="matrix14.html#T5" target="_self" title="MATRIX14:th.5">Th5</a></span>;<br/><span class="kw">hence </span><a NAME="E9:43_5_4_3_2_1_1_1"/>
<font color="Maroon" title="c10">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:43_5_4_3_2_1_1_1"><span class="lab"><font color="Green" title="E85">A103</font></span></a>, <a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</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="43_5_4_3_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:43_5_4_3_2_1_1_2"/><span class="lab"><font color="Green" title="E82">A104</font></span>: 
<font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</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="c10">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font></span><br/><div class="add"><a NAME="E2:43_5_4_3_2_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">Th23</a></span>;<br/><a NAME="E3:43_5_4_3_2_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_5_4"><span class="lab"><font color="Green" title="E73">A88</font></span></a>, <a class="txt" href="#E3:43_5_4"><span class="lab"><font color="Green" title="E74">A89</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A105</font></span>: <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font> = 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_5_4"><span class="lab"><font color="Green" title="E74">A89</font></span></a>, <a class="txt" href="#E10:43_5_4"><span class="lab"><font color="Green" title="E78">A93</font></span></a>, <a class="txt" href="#E1:43_5_4_3_2"><span class="lab"><font color="Green" title="E79">A97</font></span></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">Th25</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></span>
;<br/><a NAME="E5:43_5_4_3_2_1_1_2"/><span class="lab"><font color="Green" title="E84">A106</font></span>: 
<span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_5"><span class="lab"><font color="Green" title="E60">A75</font></span></a>, <a class="txt" href="#E2:43_5_4"><span class="lab"><font color="Green" title="E73">A88</font></span></a>, <a class="txt" href="#E3:43_5_4"><span class="lab"><font color="Green" title="E74">A89</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/><a NAME="E6:43_5_4_3_2_1_1_2"/><span class="lab"><font color="Green" title="E85">A107</font></span>: 
<font color="Maroon" title="c10">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_5_4"><span class="lab"><font color="Green" title="E73">A88</font></span></a>, <a class="txt" href="#E3:43_5_4"><span class="lab"><font color="Green" title="E74">A89</font></span></a>, <a class="txt" href="#E5:43_5_4_3_2"><span class="lab"><font color="Green" title="E81">A99</font></span></a>, <a class="txt" href="#E1:43_5_4_3_2_1_1_2"><span class="lab"><font color="Green" title="E82">A104</font></span></a>, <a class="ref" href="matrix14.html#T24" target="_self" title="MATRIX14:th.24">Th24</a></span>;<br/><span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">k</font> = 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:43_5"><span class="lab"><font color="Green" title="E69">A84</font></span></a>, <a class="txt" href="#E2:43_5_4"><span class="lab"><font color="Green" title="E73">A88</font></span></a>, <a class="txt" href="#E3:43_5_4"><span class="lab"><font color="Green" title="E74">A89</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>
<br/>.= 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_5_4"><span class="lab"><font color="Green" title="E73">A88</font></span></a>, <a class="txt" href="#E3:43_5_4"><span class="lab"><font color="Green" title="E74">A89</font></span></a>, <a class="txt" href="#E1:43_5_4_3_2_1_1_2"><span class="lab"><font color="Green" title="E82">A104</font></span></a>, <a class="ref" href="matrix14.html#T24" target="_self" title="MATRIX14:th.24">Th24</a></span>
;<br/><a NAME="E8:43_5_4_3_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_5"><span class="lab"><font color="Green" title="E60">A75</font></span></a>, <a class="txt" href="#E19:43_5"><span class="lab"><font color="Green" title="E69">A84</font></span></a>, <a class="txt" href="#E2:43_5_4"><span class="lab"><font color="Green" title="E73">A88</font></span></a>, <a class="txt" href="#E3:43_5_4"><span class="lab"><font color="Green" title="E74">A89</font></span></a>, <a class="txt" href="#E5:43_5_4_3_2_1_1_2"><span class="lab"><font color="Green" title="E84">A106</font></span></a>, <a class="txt" href="#E4:43_5_4_3_2_1_1_2"><span class="lab"><font color="Green" title="E83">A105</font></span></a>, <a class="ref" href="matrix14.html#T5" target="_self" title="MATRIX14:th.5">Th5</a></span>;<br/><span class="kw">hence </span><a NAME="E9:43_5_4_3_2_1_1_2"/>
<font color="Maroon" title="c10">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:43_5_4_3_2_1_1_2"><span class="lab"><font color="Green" title="E85">A107</font></span></a>, <a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</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><span class="kw">hence </span><a NAME="E7:43_5_4_3_2"/>
<font color="Maroon" title="c10">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font>
 ; <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 NAME="E22:43_5"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p4">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_5"><span class="lab"><font color="Green" title="E60">A75</font></span></a>, <a class="txt" href="#E19:43_5"><span class="lab"><font color="Green" title="E69">A84</font></span></a>, <a class="ref" href="matrix14.html#T2" target="_self" title="MATRIX14:th.2">Th2</a></span>;<br/>
<a NAME="E23:43_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A108</font></span>: 
 <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:43_5"><span class="lab"><font color="Green" title="E68">A83</font></span></a>, <a class="txt" href="#E21:43_5"><span class="lab"><font color="Green" title="E71">A86</font></span></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></span>;<br/>
<a NAME="E24:43_5"/><span class="lab"><font color="Green" title="E73">A109</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:43"><span class="lab"><font color="Green" title="E50">A31</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E74">A110</font></span>: <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:43"><span class="lab"><font color="Green" title="E46">A27</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>
<br/>.= 
<font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:43"><span class="lab"><font color="Green" title="E55">A70</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>
;<br/>
<span class="lab"><font color="Green" title="E75">A111</font></span>: <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">j</font> = 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_5"><span class="lab"><font color="Green" title="E57">A72</font></span></a>, <a class="txt" href="#E2:43_5"><span class="lab"><font color="Green" title="E58">A73</font></span></a>, <a class="txt" href="#E3:43_5"><span class="lab"><font color="Green" title="E59">A74</font></span></a>, <a class="txt" href="#E24:43_5"><span class="lab"><font color="Green" title="E73">A109</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>
<br/>.= 
<font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:43_5"><span class="lab"><font color="Green" title="E67">A82</font></span></a>, <a class="txt" href="#E15:43_5"><span class="lab"><font color="Green" title="E66">A81</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>
;<br/>
<a NAME="E27:43_5"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">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="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_5"><span class="lab"><font color="Green" title="E57">A72</font></span></a>, <a class="txt" href="#E2:43_5"><span class="lab"><font color="Green" title="E58">A73</font></span></a>, <a class="txt" href="#E3:43_5"><span class="lab"><font color="Green" title="E59">A74</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E76">A112</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> = 
<span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:43"><span class="lab"><font color="Green" title="E43">A10</font></span></a>, <a class="txt" href="#E15:43_5"><span class="lab"><font color="Green" title="E66">A81</font></span></a>, <a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a></span>
<br/>.= 
<span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p4">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_5"><span class="lab"><font color="Green" title="E60">A75</font></span></a>, <a class="txt" href="#E19:43_5"><span class="lab"><font color="Green" title="E69">A84</font></span></a>, <a class="txt" href="#E23:43_5"><span class="lab"><font color="Green" title="E72">A108</font></span></a>, <a class="txt" href="#E24:43_5"><span class="lab"><font color="Green" title="E73">A109</font></span></a>, <a class="ref" href="matrix14.html#T11" target="_self" title="MATRIX14:th.11">Th11</a></span>
<br/>.= 
<span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix14.html#T6" target="_self" title="MATRIX14:th.6">Th6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:43"><span class="lab"><font color="Green" title="E42">A9</font></span></a>, <a class="txt" href="#E24:43_5"><span class="lab"><font color="Green" title="E73">A109</font></span></a>, <a class="ref" href="matrix14.html#T10" target="_self" title="MATRIX14:th.10">Th10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,1</span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:43"><span class="lab"><font color="Green" title="E46">A27</font></span></a>, <a class="txt" href="#E24:43_5"><span class="lab"><font color="Green" title="E73">A109</font></span></a>, <a class="txt" href="#E25:43_5"><span class="lab"><font color="Green" title="E74">A110</font></span></a>, <a class="ref" href="matrix14.html#T35" target="_self" title="MATRIX14:th.35">Th35</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,1</span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T41" title="VECTSP_1:th.41">VECTSP_1:41</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_5"><span class="lab"><font color="Green" title="E57">A72</font></span></a>, <a class="txt" href="#E2:43_5"><span class="lab"><font color="Green" title="E58">A73</font></span></a>, <a class="txt" href="#E3:43_5"><span class="lab"><font color="Green" title="E59">A74</font></span></a>, <a class="txt" href="#E24:43_5"><span class="lab"><font color="Green" title="E73">A109</font></span></a>, <a class="txt" href="#E26:43_5"><span class="lab"><font color="Green" title="E75">A111</font></span></a>, <a class="ref" href="matrix14.html#T35" target="_self" title="MATRIX14:th.35">Th35</a></span>
;<br/>
<a NAME="E29:43_5"/><span class="lab"><font color="Green" title="E77">A113</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_5"><span class="lab"><font color="Green" title="E57">A72</font></span></a>, <a class="txt" href="#E2:43_5"><span class="lab"><font color="Green" title="E58">A73</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E30:43_5"/>
<font color="Maroon" title="c9">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:43"><span class="lab"><font color="Green" title="E40">A7</font></span></a>, <a class="txt" href="#E1:43_5"><span class="lab"><font color="Green" title="E57">A72</font></span></a>, <a class="txt" href="#E2:43_5"><span class="lab"><font color="Green" title="E58">A73</font></span></a>, <a class="txt" href="#E13:43_5"><span class="lab"><font color="Green" title="E64">A79</font></span></a></span>;<br/>
<span class="kw">hence </span><span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> = 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_5"><span class="lab"><font color="Green" title="E57">A72</font></span></a>, <a class="txt" href="#E29:43_5"><span class="lab"><font color="Green" title="E77">A113</font></span></a>, <a class="txt" href="#E28:43_5"><span class="lab"><font color="Green" title="E76">A112</font></span></a>, <a class="txt" href="#E14:43_5"><span class="lab"><font color="Green" title="E65">A80</font></span></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">Th25</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:43_5"><span class="lab"><font color="Green" title="E62">A77</font></span></a>, <a class="txt" href="#E10:43_5"><span class="lab"><font color="Green" title="E63">A78</font></span></a>, <a class="ref" href="rlvect_1.html#T25" title="RLVECT_1:th.25">RLVECT_1:25</a></span>
<br/>.= 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></span>
;<br/> <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>
<a NAME="E24:43"/><span class="lab"><font color="Green" title="E57">A114</font></span>: 
 <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<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>,<span class="p2">(<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></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></span>;<br/>
<a NAME="E25:43"/><span class="lab"><font color="Green" title="E58">A115</font></span>: 
for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">j</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_6"><span class="kw">proof </span></a><div class="add">

<span class="lab"><font color="Green" title="E59">A116</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixr1.html#T16" title="MATRIXR1:th.16">MATRIXR1:16</a></span>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">Th23</a></span>
;<br/>
<span class="kw">let </span><font color="Maroon" title="c6">i</font>, <font color="Maroon" title="c7">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">i</font> &amp; <font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">j</font> &amp; <font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c6">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">j</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:43_6"/><span class="lab"><font color="Green" title="E60">A117</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">i</font>
 <span class="kw">and </span><br/><a NAME="E3:43_6"/><span class="lab"><font color="Green" title="E61">A118</font></span>: 
<font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">and </span><br/><a NAME="E4:43_6"/><span class="lab"><font color="Green" title="E62">A119</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">j</font>
 <span class="kw">and </span><br/><a NAME="E5:43_6"/><span class="lab"><font color="Green" title="E63">A120</font></span>: 
<font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">and </span><br/><a NAME="E6:43_6"/><span class="lab"><font color="Green" title="E64">A121</font></span>: 
<font color="Maroon" title="c6">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">j</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="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span><br/>

<a NAME="E7:43_6"/><span class="lab"><font color="Green" title="E65">A122</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">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="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_6"><span class="lab"><font color="Green" title="E60">A117</font></span></a>, <a class="txt" href="#E3:43_6"><span class="lab"><font color="Green" title="E61">A118</font></span></a>, <a class="txt" href="#E4:43_6"><span class="lab"><font color="Green" title="E62">A119</font></span></a>, <a class="txt" href="#E5:43_6"><span class="lab"><font color="Green" title="E63">A120</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E8:43_6"/><span class="lab"><font color="Green" title="E66">A123</font></span>: 
<font color="Maroon" title="c7">j</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> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_6"><span class="lab"><font color="Green" title="E62">A119</font></span></a>, <a class="txt" href="#E5:43_6"><span class="lab"><font color="Green" title="E63">A120</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<span class="lab"><font color="Green" title="E67">A124</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixr1.html#T16" title="MATRIXR1:th.16">MATRIXR1:16</a></span>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">Th23</a></span>
;<br/>
<span class="lab"><font color="Green" title="E68">A125</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c7">j</font></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">P</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></span>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></span>
;<br/>
<a NAME="E11:43_6"/><span class="lab"><font color="Green" title="E69">A126</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">i</font>,1</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> <font color="Maroon" title="c5">Q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:43"><span class="lab"><font color="Green" title="E46">A27</font></span></a>, <a class="txt" href="#E2:43_6"><span class="lab"><font color="Green" title="E60">A117</font></span></a>, <a class="txt" href="#E3:43_6"><span class="lab"><font color="Green" title="E61">A118</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E12:43_6"/><span class="lab"><font color="Green" title="E70">A127</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">Th23</a></span>;<br/>
<span class="lab"><font color="Green" title="E71">A128</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix14.html#T6" target="_self" title="MATRIX14:th.6">Th6</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixr1.html#T16" title="MATRIXR1:th.16">MATRIXR1:16</a></span>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">Th23</a></span>
;<br/>
<a NAME="E14:43_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A129</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p4">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:43_6"><span class="lab"><font color="Green" title="E70">A127</font></span></a>, <a class="ref" href="matrix14.html#T2" target="_self" title="MATRIX14:th.2">Th2</a></span>;<br/>
<a NAME="E15:43_6"/><span class="lab"><font color="Green" title="E73">A130</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">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> <font color="Maroon" title="c5">Q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_6"><span class="lab"><font color="Green" title="E60">A117</font></span></a>, <a class="txt" href="#E3:43_6"><span class="lab"><font color="Green" title="E61">A118</font></span></a>, <a class="txt" href="#E4:43_6"><span class="lab"><font color="Green" title="E62">A119</font></span></a>, <a class="txt" href="#E5:43_6"><span class="lab"><font color="Green" title="E63">A120</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_6_5"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_6_5_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="E1:43_6_5_1"/>
( <font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 or <font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_6_5_1_1"><span class="kw">suppose </span></a><a NAME="E1:43_6_5_1_1"/><span class="lab"><font color="Green" title="E74">A131</font></span>: 
<font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ; <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="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c8">p</font> =  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c7">j</font>, <font color="Maroon" title="c9">q</font> = <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font> ;<br/><a NAME="E3:43_6_5_1_1"/>
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> <font color="Maroon" title="c1">n</font> holds <br/><font color="Maroon" title="c8">p</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="c9">q</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="43_6_5_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">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="c10">k</font> &amp; <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> implies <font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:43_6_5_1_1_1"/><span class="lab"><font color="Green" title="E75">A132</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font>
 <span class="kw">and </span><br/><a NAME="E2:43_6_5_1_1_1"/><span class="lab"><font color="Green" title="E76">A133</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</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="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/>

<a NAME="E3:43_6_5_1_1_1"/>
<font color="Maroon" title="c10">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> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_6_5_1_1_1"><span class="lab"><font color="Green" title="E75">A132</font></span></a>, <a class="txt" href="#E2:43_6_5_1_1_1"><span class="lab"><font color="Green" title="E76">A133</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E4:43_6_5_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A134</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:43_6"><span class="lab"><font color="Green" title="E67">A124</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E5:43_6_5_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">Th23</a></span>;<br/>
<a NAME="E6:43_6_5_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A135</font></span>: 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_6_5_1_1_1"><span class="lab"><font color="Green" title="E75">A132</font></span></a>, <a class="txt" href="#E2:43_6_5_1_1_1"><span class="lab"><font color="Green" title="E76">A133</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/>
<a NAME="E7:43_6_5_1_1_1"/>
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:43"><span class="lab"><font color="Green" title="E43">A10</font></span></a>, <a class="txt" href="#E1:43_6_5_1_1_1"><span class="lab"><font color="Green" title="E75">A132</font></span></a>, <a class="txt" href="#E2:43_6_5_1_1_1"><span class="lab"><font color="Green" title="E76">A133</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E8:43_6_5_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A136</font></span>: 
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c7">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></span>;<br/>
<span class="lab"><font color="Green" title="E80">A137</font></span>: <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix14.html#T6" target="_self" title="MATRIX14:th.6">Th6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_6_5_1_1_1"><span class="lab"><font color="Green" title="E77">A134</font></span></a>, <a class="txt" href="#E6:43_6_5_1_1_1"><span class="lab"><font color="Green" title="E78">A135</font></span></a>, <a class="ref" href="fvsum_1.html#T62" title="FVSUM_1:th.62">FVSUM_1:62</a></span>
;<br/>
<a NAME="E10:43_6_5_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">Th23</a></span>;<br/>
<a NAME="E11:43_6_5_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A138</font></span>: 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_6_5_1_1_1"><span class="lab"><font color="Green" title="E75">A132</font></span></a>, <a class="txt" href="#E2:43_6_5_1_1_1"><span class="lab"><font color="Green" title="E76">A133</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_6_5_1_1_1_2"><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="E1:43_6_5_1_1_1_2"/>
( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</font> or 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">k</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_6_5_1_1_1"><span class="lab"><font color="Green" title="E75">A132</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_6_5_1_1_1_2_1"><span class="kw">suppose </span></a><a NAME="E1:43_6_5_1_1_1_2_1"/><span class="lab"><font color="Green" title="E82">A139</font></span>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</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="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/><div class="add"><a NAME="E2:43_6_5_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A140</font></span>: 
<span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:43"><span class="lab"><font color="Green" title="E46">A27</font></span></a>, <a class="txt" href="#E6:43_6_5_1_1_1"><span class="lab"><font color="Green" title="E78">A135</font></span></a>, <a class="txt" href="#E9:43_6_5_1_1_1"><span class="lab"><font color="Green" title="E80">A137</font></span></a>, <a class="ref" href="matrix14.html#T24" target="_self" title="MATRIX14:th.24">Th24</a></span>;<br/><a NAME="E3:43_6_5_1_1_1_2_1"/><span class="lab"><font color="Green" title="E84">A141</font></span>: 
<span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:43_6"><span class="lab"><font color="Green" title="E71">A128</font></span></a>, <a class="txt" href="#E1:43_6_5_1_1_1"><span class="lab"><font color="Green" title="E75">A132</font></span></a>, <a class="txt" href="#E2:43_6_5_1_1_1"><span class="lab"><font color="Green" title="E76">A133</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/><span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:43_6"><span class="lab"><font color="Green" title="E70">A127</font></span></a>, <a class="txt" href="#E1:43_6_5_1_1_1"><span class="lab"><font color="Green" title="E75">A132</font></span></a>, <a class="txt" href="#E2:43_6_5_1_1_1"><span class="lab"><font color="Green" title="E76">A133</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_6_5_1_1"><span class="lab"><font color="Green" title="E74">A131</font></span></a>, <a class="txt" href="#E2:43_6_5_1_1_1"><span class="lab"><font color="Green" title="E76">A133</font></span></a>, <a class="txt" href="#E1:43_6_5_1_1_1_2_1"><span class="lab"><font color="Green" title="E82">A139</font></span></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">Th25</a></span>
;<br/><span class="kw">then </span><font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:43_6"><span class="lab"><font color="Green" title="E71">A128</font></span></a>, <a class="txt" href="#E12:43_6"><span class="lab"><font color="Green" title="E70">A127</font></span></a>, <a class="txt" href="#E1:43_6_5_1_1_1"><span class="lab"><font color="Green" title="E75">A132</font></span></a>, <a class="txt" href="#E2:43_6_5_1_1_1"><span class="lab"><font color="Green" title="E76">A133</font></span></a>, <a class="txt" href="#E3:43_6_5_1_1_1_2_1"><span class="lab"><font color="Green" title="E84">A141</font></span></a>, <a class="txt" href="#E2:43_6_5_1_1_1_2_1"><span class="lab"><font color="Green" title="E83">A140</font></span></a>, <a class="ref" href="matrix14.html#T5" target="_self" title="MATRIX14:th.5">Th5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></span>
<br/>.= 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D16" title="VECTSP_1:def.16">VECTSP_1:def 16</a></span>
;<br/><span class="kw">hence </span><a NAME="E6:43_6_5_1_1_1_2_1"/>
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:43"><span class="lab"><font color="Green" title="E53">A34</font></span></a>, <a class="txt" href="#E5:43_6"><span class="lab"><font color="Green" title="E63">A120</font></span></a>, <a class="txt" href="#E1:43_6_5_1_1"><span class="lab"><font color="Green" title="E74">A131</font></span></a>, <a class="txt" href="#E8:43_6_5_1_1_1"><span class="lab"><font color="Green" title="E79">A136</font></span></a>, <a class="txt" href="#E1:43_6_5_1_1_1_2_1"><span class="lab"><font color="Green" title="E82">A139</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_6_5_1_1_1_2_2"><span class="kw">suppose </span></a><a NAME="E1:43_6_5_1_1_1_2_2"/><span class="lab"><font color="Green" title="E82">A142</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">k</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="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/><div class="add"><a NAME="E2:43_6_5_1_1_1_2_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c7">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> <font color="Maroon" title="c4">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_6"><span class="lab"><font color="Green" title="E62">A119</font></span></a>, <a class="txt" href="#E5:43_6"><span class="lab"><font color="Green" title="E63">A120</font></span></a>, <a class="txt" href="#E1:43_6_5_1_1_1"><span class="lab"><font color="Green" title="E75">A132</font></span></a>, <a class="txt" href="#E2:43_6_5_1_1_1"><span class="lab"><font color="Green" title="E76">A133</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E3:43_6_5_1_1_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A143</font></span>: 
 ex <font color="Olive" title="b1">p2</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font> st <br/>( <font color="Olive" title="b1">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> &amp; <font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</a></span>;<br/><a NAME="E4:43_6_5_1_1_1_2_2"/>
<font color="Maroon" title="c10">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="E5:43_6_5_1_1_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A144</font></span>: 
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c10">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43"><span class="lab"><font color="Green" title="E37">A4</font></span></a>, <a class="txt" href="#E2:43_6_5_1_1_1"><span class="lab"><font color="Green" title="E76">A133</font></span></a>, <a class="txt" href="#E8:43_6_5_1_1_1"><span class="lab"><font color="Green" title="E79">A136</font></span></a>, <a class="txt" href="#E1:43_6_5_1_1_1_2_2"><span class="lab"><font color="Green" title="E82">A142</font></span></a>, <a class="txt" href="#E3:43_6_5_1_1_1_2_2"><span class="lab"><font color="Green" title="E83">A143</font></span></a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_6_5_1_1_1_2_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_6_5_1_1_1_2_2_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="E1:43_6_5_1_1_1_2_2_1_1"/>
( <font color="Maroon" title="c10">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">j</font> or <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_6_5_1_1_1_2_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:43_6_5_1_1_1_2_2_1_1_1"/><span class="lab"><font color="Green" title="E85">A145</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">j</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="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/><div class="add"><span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:43_6"><span class="lab"><font color="Green" title="E71">A128</font></span></a>, <a class="txt" href="#E1:43_6_5_1_1_1"><span class="lab"><font color="Green" title="E75">A132</font></span></a>, <a class="txt" href="#E2:43_6_5_1_1_1"><span class="lab"><font color="Green" title="E76">A133</font></span></a>, <a class="txt" href="#E9:43_6_5_1_1_1"><span class="lab"><font color="Green" title="E80">A137</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_6_5_1_1_1"><span class="lab"><font color="Green" title="E76">A133</font></span></a>, <a class="txt" href="#E11:43_6_5_1_1_1"><span class="lab"><font color="Green" title="E81">A138</font></span></a>, <a class="txt" href="#E1:43_6_5_1_1_1_2_2"><span class="lab"><font color="Green" title="E82">A142</font></span></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">Th25</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></span>
;<br/><span class="kw">then </span><font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</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"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:43_6"><span class="lab"><font color="Green" title="E71">A128</font></span></a>, <a class="txt" href="#E12:43_6"><span class="lab"><font color="Green" title="E70">A127</font></span></a>, <a class="txt" href="#E1:43_6_5_1_1_1"><span class="lab"><font color="Green" title="E75">A132</font></span></a>, <a class="txt" href="#E2:43_6_5_1_1_1"><span class="lab"><font color="Green" title="E76">A133</font></span></a>, <a class="ref" href="matrix14.html#T5" target="_self" title="MATRIX14:th.5">Th5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:43_6"><span class="lab"><font color="Green" title="E70">A127</font></span></a>, <a class="txt" href="#E1:43_6_5_1_1_1"><span class="lab"><font color="Green" title="E75">A132</font></span></a>, <a class="txt" href="#E2:43_6_5_1_1_1"><span class="lab"><font color="Green" title="E76">A133</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_6_5_1_1_1"><span class="lab"><font color="Green" title="E75">A132</font></span></a>, <a class="txt" href="#E2:43_6_5_1_1_1"><span class="lab"><font color="Green" title="E76">A133</font></span></a>, <a class="txt" href="#E1:43_6_5_1_1_1_2_2_1_1_1"><span class="lab"><font color="Green" title="E85">A145</font></span></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">Th25</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:43_6_5_1_1_1_2_2_1_1_1"/>
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_6"><span class="lab"><font color="Green" title="E62">A119</font></span></a>, <a class="txt" href="#E5:43_6"><span class="lab"><font color="Green" title="E63">A120</font></span></a>, <a class="txt" href="#E5:43_6_5_1_1_1_2_2"><span class="lab"><font color="Green" title="E84">A144</font></span></a>, <a class="txt" href="#E1:43_6_5_1_1_1_2_2_1_1_1"><span class="lab"><font color="Green" title="E85">A145</font></span></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">Th25</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="43_6_5_1_1_1_2_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:43_6_5_1_1_1_2_2_1_1_2"/><span class="lab"><font color="Green" title="E85">A146</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</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="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/><div class="add"><span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:43_6"><span class="lab"><font color="Green" title="E71">A128</font></span></a>, <a class="txt" href="#E1:43_6_5_1_1_1"><span class="lab"><font color="Green" title="E75">A132</font></span></a>, <a class="txt" href="#E2:43_6_5_1_1_1"><span class="lab"><font color="Green" title="E76">A133</font></span></a>, <a class="txt" href="#E9:43_6_5_1_1_1"><span class="lab"><font color="Green" title="E80">A137</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_6_5_1_1_1"><span class="lab"><font color="Green" title="E76">A133</font></span></a>, <a class="txt" href="#E11:43_6_5_1_1_1"><span class="lab"><font color="Green" title="E81">A138</font></span></a>, <a class="txt" href="#E1:43_6_5_1_1_1_2_2"><span class="lab"><font color="Green" title="E82">A142</font></span></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">Th25</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></span>
;<br/><span class="kw">then </span><font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</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"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:43_6"><span class="lab"><font color="Green" title="E71">A128</font></span></a>, <a class="txt" href="#E12:43_6"><span class="lab"><font color="Green" title="E70">A127</font></span></a>, <a class="txt" href="#E1:43_6_5_1_1_1"><span class="lab"><font color="Green" title="E75">A132</font></span></a>, <a class="txt" href="#E2:43_6_5_1_1_1"><span class="lab"><font color="Green" title="E76">A133</font></span></a>, <a class="ref" href="matrix14.html#T5" target="_self" title="MATRIX14:th.5">Th5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:43_6"><span class="lab"><font color="Green" title="E70">A127</font></span></a>, <a class="txt" href="#E1:43_6_5_1_1_1"><span class="lab"><font color="Green" title="E75">A132</font></span></a>, <a class="txt" href="#E2:43_6_5_1_1_1"><span class="lab"><font color="Green" title="E76">A133</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>
<br/>.= 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_6"><span class="lab"><font color="Green" title="E62">A119</font></span></a>, <a class="txt" href="#E5:43_6"><span class="lab"><font color="Green" title="E63">A120</font></span></a>, <a class="txt" href="#E1:43_6_5_1_1_1_2_2_1_1_2"><span class="lab"><font color="Green" title="E85">A146</font></span></a>, <a class="ref" href="matrix14.html#T24" target="_self" title="MATRIX14:th.24">Th24</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:43_6_5_1_1_1_2_2_1_1_2"/>
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_6_5_1_1_1"><span class="lab"><font color="Green" title="E75">A132</font></span></a>, <a class="txt" href="#E2:43_6_5_1_1_1"><span class="lab"><font color="Green" title="E76">A133</font></span></a>, <a class="txt" href="#E5:43_6_5_1_1_1_2_2"><span class="lab"><font color="Green" title="E84">A144</font></span></a>, <a class="txt" href="#E1:43_6_5_1_1_1_2_2_1_1_2"><span class="lab"><font color="Green" title="E85">A146</font></span></a>, <a class="ref" href="matrix14.html#T24" target="_self" title="MATRIX14:th.24">Th24</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><span class="kw">hence </span><a NAME="E7:43_6_5_1_1_1_2_2"/>
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 ; <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 NAME="E4:43_6_5_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A147</font></span>: 
 <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:43_6"><span class="lab"><font color="Green" title="E68">A125</font></span></a>, <a class="txt" href="#E14:43_6"><span class="lab"><font color="Green" title="E72">A129</font></span></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></span>;<br/><a NAME="E5:43_6_5_1_1"/><span class="lab"><font color="Green" title="E76">A148</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_6"><span class="lab"><font color="Green" title="E60">A117</font></span></a>, <a class="txt" href="#E3:43_6"><span class="lab"><font color="Green" title="E61">A118</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E6:43_6_5_1_1"/><span class="lab"><font color="Green" title="E77">A149</font></span>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></span>;<br/><a NAME="E7:43_6_5_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A150</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A151</font></span>: <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">j</font> = 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_6"><span class="lab"><font color="Green" title="E62">A119</font></span></a>, <a class="txt" href="#E5:43_6"><span class="lab"><font color="Green" title="E63">A120</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>
<br/>.= 
<font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:43_6"><span class="lab"><font color="Green" title="E66">A123</font></span></a>, <a class="txt" href="#E6:43_6_5_1_1"><span class="lab"><font color="Green" title="E77">A149</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>
;<br/><span class="lab"><font color="Green" title="E80">A152</font></span>: <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:43"><span class="lab"><font color="Green" title="E46">A27</font></span></a>, <a class="txt" href="#E7:43_6_5_1_1"><span class="lab"><font color="Green" title="E78">A150</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>
<br/>.= 
<font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:43"><span class="lab"><font color="Green" title="E55">A70</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>
;<br/><span class="lab"><font color="Green" title="E81">A153</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> = 
<span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:43"><span class="lab"><font color="Green" title="E43">A10</font></span></a>, <a class="txt" href="#E7:43_6"><span class="lab"><font color="Green" title="E65">A122</font></span></a>, <a class="txt" href="#E6:43_6_5_1_1"><span class="lab"><font color="Green" title="E77">A149</font></span></a>, <a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a></span>
<br/>.= 
<span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p4">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:43_6"><span class="lab"><font color="Green" title="E71">A128</font></span></a>, <a class="txt" href="#E12:43_6"><span class="lab"><font color="Green" title="E70">A127</font></span></a>, <a class="txt" href="#E4:43_6_5_1_1"><span class="lab"><font color="Green" title="E75">A147</font></span></a>, <a class="txt" href="#E7:43_6_5_1_1"><span class="lab"><font color="Green" title="E78">A150</font></span></a>, <a class="ref" href="matrix14.html#T11" target="_self" title="MATRIX14:th.11">Th11</a></span>
<br/>.= 
<span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix14.html#T6" target="_self" title="MATRIX14:th.6">Th6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:43"><span class="lab"><font color="Green" title="E42">A9</font></span></a>, <a class="txt" href="#E7:43_6_5_1_1"><span class="lab"><font color="Green" title="E78">A150</font></span></a>, <a class="ref" href="matrix14.html#T10" target="_self" title="MATRIX14:th.10">Th10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,1</span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:43"><span class="lab"><font color="Green" title="E46">A27</font></span></a>, <a class="txt" href="#E7:43_6_5_1_1"><span class="lab"><font color="Green" title="E78">A150</font></span></a>, <a class="txt" href="#E9:43_6_5_1_1"><span class="lab"><font color="Green" title="E80">A152</font></span></a>, <a class="ref" href="matrix14.html#T35" target="_self" title="MATRIX14:th.35">Th35</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,1</span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T41" title="VECTSP_1:th.41">VECTSP_1:41</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_6"><span class="lab"><font color="Green" title="E62">A119</font></span></a>, <a class="txt" href="#E5:43_6"><span class="lab"><font color="Green" title="E63">A120</font></span></a>, <a class="txt" href="#E7:43_6_5_1_1"><span class="lab"><font color="Green" title="E78">A150</font></span></a>, <a class="txt" href="#E8:43_6_5_1_1"><span class="lab"><font color="Green" title="E79">A151</font></span></a>, <a class="ref" href="matrix14.html#T35" target="_self" title="MATRIX14:th.35">Th35</a></span>
;<br/><span class="kw">consider </span><font color="Maroon" title="c10">p2</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font><span class="kw"> such that </span><br/><a NAME="E12:43_6_5_1_1"/><span class="lab"><font color="Green" title="E82">A154</font></span>: 
<font color="Maroon" title="c10">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
 <span class="kw">and </span><br/><a NAME="E13:43_6_5_1_1"/><span class="lab"><font color="Green" title="E83">A155</font></span>: 
<font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:43_6"><span class="lab"><font color="Green" title="E69">A126</font></span></a>, <a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c11">p1</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font><span class="kw"> such that </span><br/><a NAME="E15:43_6_5_1_1"/><span class="lab"><font color="Green" title="E84">A156</font></span>: 
<font color="Maroon" title="c11">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
 <span class="kw">and </span><br/><a NAME="E16:43_6_5_1_1"/><span class="lab"><font color="Green" title="E85">A157</font></span>: 
<font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:43_6"><span class="lab"><font color="Green" title="E73">A130</font></span></a>, <a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</a></span>;<br/><a NAME="E17:43_6_5_1_1"/>
<font color="Maroon" title="c11">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:43"><span class="lab"><font color="Green" title="E40">A7</font></span></a>, <a class="txt" href="#E2:43_6"><span class="lab"><font color="Green" title="E60">A117</font></span></a>, <a class="txt" href="#E3:43_6"><span class="lab"><font color="Green" title="E61">A118</font></span></a>, <a class="txt" href="#E15:43_6_5_1_1"><span class="lab"><font color="Green" title="E84">A156</font></span></a></span>;<br/><a NAME="E18:43_6_5_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A158</font></span>: 
<font color="Maroon" title="c11">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_6"><span class="lab"><font color="Green" title="E62">A119</font></span></a>, <a class="txt" href="#E5:43_6"><span class="lab"><font color="Green" title="E63">A120</font></span></a>, <a class="txt" href="#E6:43_6"><span class="lab"><font color="Green" title="E64">A121</font></span></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">Th25</a></span>;<br/><a NAME="E19:43_6_5_1_1"/>
<font color="Maroon" title="c10">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:43"><span class="lab"><font color="Green" title="E40">A7</font></span></a>, <a class="txt" href="#E2:43_6"><span class="lab"><font color="Green" title="E60">A117</font></span></a>, <a class="txt" href="#E3:43_6"><span class="lab"><font color="Green" title="E61">A118</font></span></a>, <a class="txt" href="#E12:43_6_5_1_1"><span class="lab"><font color="Green" title="E82">A154</font></span></a></span>;<br/><span class="kw">hence </span><span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> = 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_6"><span class="lab"><font color="Green" title="E60">A117</font></span></a>, <a class="txt" href="#E5:43_6_5_1_1"><span class="lab"><font color="Green" title="E76">A148</font></span></a>, <a class="txt" href="#E10:43_6_5_1_1"><span class="lab"><font color="Green" title="E81">A153</font></span></a>, <a class="txt" href="#E13:43_6_5_1_1"><span class="lab"><font color="Green" title="E83">A155</font></span></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">Th25</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:43_6_5_1_1"><span class="lab"><font color="Green" title="E85">A157</font></span></a>, <a class="txt" href="#E18:43_6_5_1_1"><span class="lab"><font color="Green" title="E86">A158</font></span></a>, <a class="ref" href="rlvect_1.html#T25" title="RLVECT_1:th.25">RLVECT_1:25</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></span>
;<br/> <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="43_6_5_1_2"><span class="kw">suppose </span></a><a NAME="E1:43_6_5_1_2"/><span class="lab"><font color="Green" title="E74">A159</font></span>: 
<font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 ; <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="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c8">p</font> =  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c7">j</font>, <font color="Maroon" title="c9">q</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font> ;<br/><a NAME="E3:43_6_5_1_2"/><span class="lab"><font color="Green" title="E75">A160</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> <font color="Maroon" title="c1">n</font> holds <br/><font color="Maroon" title="c8">p</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="c9">q</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="43_6_5_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">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="c10">k</font> &amp; <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> implies <font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:43_6_5_1_2_1"/><span class="lab"><font color="Green" title="E76">A161</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font>
 <span class="kw">and </span><br/><a NAME="E2:43_6_5_1_2_1"/><span class="lab"><font color="Green" title="E77">A162</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</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="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/>

<a NAME="E3:43_6_5_1_2_1"/><span class="lab"><font color="Green" title="E78">A163</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">Th23</a></span>;<br/>
<a NAME="E4:43_6_5_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A164</font></span>: 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_6_5_1_2_1"><span class="lab"><font color="Green" title="E76">A161</font></span></a>, <a class="txt" href="#E2:43_6_5_1_2_1"><span class="lab"><font color="Green" title="E77">A162</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/>
<a NAME="E5:43_6_5_1_2_1"/><span class="lab"><font color="Green" title="E80">A165</font></span>: 
<font color="Maroon" title="c10">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> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_6_5_1_2_1"><span class="lab"><font color="Green" title="E76">A161</font></span></a>, <a class="txt" href="#E2:43_6_5_1_2_1"><span class="lab"><font color="Green" title="E77">A162</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E6:43_6_5_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_6"><span class="lab"><font color="Green" title="E59">A116</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E7:43_6_5_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A166</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_6_5_1_2_1"><span class="lab"><font color="Green" title="E79">A164</font></span></a>, <a class="ref" href="fvsum_1.html#T62" title="FVSUM_1:th.62">FVSUM_1:62</a></span>;<br/>
<a NAME="E8:43_6_5_1_2_1"/>
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:43"><span class="lab"><font color="Green" title="E43">A10</font></span></a>, <a class="txt" href="#E1:43_6_5_1_2_1"><span class="lab"><font color="Green" title="E76">A161</font></span></a>, <a class="txt" href="#E2:43_6_5_1_2_1"><span class="lab"><font color="Green" title="E77">A162</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E9:43_6_5_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E82">A167</font></span>: 
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c7">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_6_5_1_2_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="E1:43_6_5_1_2_1_1"/>
( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</font> or 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">k</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_6_5_1_2_1"><span class="lab"><font color="Green" title="E76">A161</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_6_5_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:43_6_5_1_2_1_1_1"/><span class="lab"><font color="Green" title="E83">A168</font></span>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</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="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/><div class="add"><span class="kw">then </span><font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_6_5_1_2_1"><span class="lab"><font color="Green" title="E77">A162</font></span></a>, <a class="txt" href="#E4:43_6_5_1_2_1"><span class="lab"><font color="Green" title="E79">A164</font></span></a>, <a class="txt" href="#E7:43_6_5_1_2_1"><span class="lab"><font color="Green" title="E81">A166</font></span></a>, <a class="ref" href="matrix14.html#T24" target="_self" title="MATRIX14:th.24">Th24</a></span>
<br/>.= 
<font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D16" title="VECTSP_1:def.16">VECTSP_1:def 16</a></span>
;<br/><span class="kw">hence </span><a NAME="E3:43_6_5_1_2_1_1_1"/>
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E36">A3</font></span></a>, <a class="txt" href="#E4:43_6"><span class="lab"><font color="Green" title="E62">A119</font></span></a>, <a class="txt" href="#E1:43_6_5_1_2"><span class="lab"><font color="Green" title="E74">A159</font></span></a>, <a class="txt" href="#E9:43_6_5_1_2_1"><span class="lab"><font color="Green" title="E82">A167</font></span></a>, <a class="txt" href="#E1:43_6_5_1_2_1_1_1"><span class="lab"><font color="Green" title="E83">A168</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</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="43_6_5_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:43_6_5_1_2_1_1_2"/><span class="lab"><font color="Green" title="E83">A169</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">k</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="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/><div class="add"><a NAME="E2:43_6_5_1_2_1_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c7">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> <font color="Maroon" title="c4">P</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:43"><span class="lab"><font color="Green" title="E57">A114</font></span></a>, <a class="txt" href="#E8:43_6"><span class="lab"><font color="Green" title="E66">A123</font></span></a>, <a class="txt" href="#E5:43_6_5_1_2_1"><span class="lab"><font color="Green" title="E80">A165</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E3:43_6_5_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A170</font></span>: 
 ex <font color="Olive" title="b1">p2</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font> st <br/>( <font color="Olive" title="b1">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> &amp; <font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</a></span>;<br/><a NAME="E4:43_6_5_1_2_1_1_2"/>
<font color="Maroon" title="c10">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="E5:43_6_5_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A171</font></span>: 
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c10">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43"><span class="lab"><font color="Green" title="E37">A4</font></span></a>, <a class="txt" href="#E2:43_6_5_1_2_1"><span class="lab"><font color="Green" title="E77">A162</font></span></a>, <a class="txt" href="#E9:43_6_5_1_2_1"><span class="lab"><font color="Green" title="E82">A167</font></span></a>, <a class="txt" href="#E1:43_6_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E83">A169</font></span></a>, <a class="txt" href="#E3:43_6_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E84">A170</font></span></a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_6_5_1_2_1_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_6_5_1_2_1_1_2_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="E1:43_6_5_1_2_1_1_2_1_1"/>
( <font color="Maroon" title="c10">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">j</font> or <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_6_5_1_2_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:43_6_5_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E86">A172</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">j</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="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/><div class="add"><a NAME="E2:43_6_5_1_2_1_1_2_1_1_1"/>
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_6_5_1_2_1"><span class="lab"><font color="Green" title="E76">A161</font></span></a>, <a class="txt" href="#E2:43_6_5_1_2_1"><span class="lab"><font color="Green" title="E77">A162</font></span></a>, <a class="txt" href="#E3:43_6_5_1_2_1"><span class="lab"><font color="Green" title="E78">A163</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/><span class="kw">then </span><span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_6_5_1_2_1"><span class="lab"><font color="Green" title="E77">A162</font></span></a>, <a class="txt" href="#E1:43_6_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E83">A169</font></span></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">Th25</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:43_6_5_1_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_6"><span class="lab"><font color="Green" title="E62">A119</font></span></a>, <a class="txt" href="#E5:43_6"><span class="lab"><font color="Green" title="E63">A120</font></span></a>, <a class="txt" href="#E7:43_6_5_1_2_1"><span class="lab"><font color="Green" title="E81">A166</font></span></a>, <a class="txt" href="#E5:43_6_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E85">A171</font></span></a>, <a class="txt" href="#E1:43_6_5_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E86">A172</font></span></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">Th25</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="43_6_5_1_2_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:43_6_5_1_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</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="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:43_6_5_1_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_6_5_1_2"><span class="lab"><font color="Green" title="E74">A159</font></span></a>, <a class="txt" href="#E1:43_6_5_1_2_1_1_2"><span class="lab"><font color="Green" title="E83">A169</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E7:43_6_5_1_2_1_1_2"/>
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 ; <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 NAME="E4:43_6_5_1_2"/><span class="lab"><font color="Green" title="E76">A173</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_6"><span class="lab"><font color="Green" title="E60">A117</font></span></a>, <a class="txt" href="#E3:43_6"><span class="lab"><font color="Green" title="E61">A118</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E5:43_6_5_1_2"/><span class="lab"><font color="Green" title="E77">A174</font></span>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></span>;<br/><a NAME="E6:43_6_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A175</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A176</font></span>: <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:43"><span class="lab"><font color="Green" title="E46">A27</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>
<br/>.= 
<font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:43"><span class="lab"><font color="Green" title="E55">A70</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>
;<br/><span class="lab"><font color="Green" title="E80">A177</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> = 
<span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:43"><span class="lab"><font color="Green" title="E43">A10</font></span></a>, <a class="txt" href="#E7:43_6"><span class="lab"><font color="Green" title="E65">A122</font></span></a>, <a class="txt" href="#E5:43_6_5_1_2"><span class="lab"><font color="Green" title="E77">A174</font></span></a>, <a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a></span>
<br/>.= 
<span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:43_6"><span class="lab"><font color="Green" title="E68">A125</font></span></a>, <a class="txt" href="#E1:43_6"><span class="lab"><font color="Green" title="E59">A116</font></span></a>, <a class="txt" href="#E3:43_6_5_1_2"><span class="lab"><font color="Green" title="E75">A160</font></span></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:43"><span class="lab"><font color="Green" title="E42">A9</font></span></a>, <a class="txt" href="#E6:43_6_5_1_2"><span class="lab"><font color="Green" title="E78">A175</font></span></a>, <a class="ref" href="matrix14.html#T10" target="_self" title="MATRIX14:th.10">Th10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:43"><span class="lab"><font color="Green" title="E46">A27</font></span></a>, <a class="txt" href="#E6:43_6_5_1_2"><span class="lab"><font color="Green" title="E78">A175</font></span></a>, <a class="txt" href="#E7:43_6_5_1_2"><span class="lab"><font color="Green" title="E79">A176</font></span></a>, <a class="ref" href="matrix14.html#T35" target="_self" title="MATRIX14:th.35">Th35</a></span>
;<br/><span class="kw">consider </span><font color="Maroon" title="c10">p2</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font><span class="kw"> such that </span><br/><a NAME="E10:43_6_5_1_2"/><span class="lab"><font color="Green" title="E81">A178</font></span>: 
<font color="Maroon" title="c10">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
 <span class="kw">and </span><br/><a NAME="E11:43_6_5_1_2"/><span class="lab"><font color="Green" title="E82">A179</font></span>: 
<font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:43_6"><span class="lab"><font color="Green" title="E69">A126</font></span></a>, <a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</a></span>;<br/><a NAME="E12:43_6_5_1_2"/>
<font color="Maroon" title="c10">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:43"><span class="lab"><font color="Green" title="E40">A7</font></span></a>, <a class="txt" href="#E2:43_6"><span class="lab"><font color="Green" title="E60">A117</font></span></a>, <a class="txt" href="#E3:43_6"><span class="lab"><font color="Green" title="E61">A118</font></span></a>, <a class="txt" href="#E10:43_6_5_1_2"><span class="lab"><font color="Green" title="E81">A178</font></span></a></span>;<br/><span class="kw">hence </span><span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_6"><span class="lab"><font color="Green" title="E60">A117</font></span></a>, <a class="txt" href="#E4:43_6_5_1_2"><span class="lab"><font color="Green" title="E76">A173</font></span></a>, <a class="txt" href="#E8:43_6_5_1_2"><span class="lab"><font color="Green" title="E80">A177</font></span></a>, <a class="txt" href="#E11:43_6_5_1_2"><span class="lab"><font color="Green" title="E82">A179</font></span></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">Th25</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></span>
;<br/> <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>
<span class="kw">hence </span><a NAME="E17:43_6"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <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"> verum</span><br/>


</div><span class="kw">end;</span></div>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c4">P</font>,1</span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">P</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></span>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></span>
;<br/>
<a NAME="E27:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A180</font></span>: 
 <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c4">P</font>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:43"><span class="lab"><font color="Green" title="E44">A11</font></span></a>, <a class="txt" href="#E12:43"><span class="lab"><font color="Green" title="E45">A12</font></span></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></span>;<br/>
<a NAME="E28:43"/><span class="lab"><font color="Green" title="E60">A181</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="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<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>,<span class="p2">(<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></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></span>;<br/>
<span class="lab"><font color="Green" title="E61">A182</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixr1.html#T16" title="MATRIXR1:th.16">MATRIXR1:16</a></span>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:43"><span class="lab"><font color="Green" title="E50">A31</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>
;<br/>
<a NAME="E30:43"/>
(  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<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>,<span class="p2">(<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></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,1</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> <font color="Maroon" title="c5">Q</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:43"><span class="lab"><font color="Green" title="E46">A27</font></span></a>, <a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E62">A183</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1 = 
<span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c4">P</font>,1</span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:43"><span class="lab"><font color="Green" title="E43">A10</font></span></a>, <a class="txt" href="#E17:43"><span class="lab"><font color="Green" title="E50">A31</font></span></a>, <a class="txt" href="#E28:43"><span class="lab"><font color="Green" title="E60">A181</font></span></a>, <a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:43"><span class="lab"><font color="Green" title="E51">A32</font></span></a>, <a class="txt" href="#E9:43"><span class="lab"><font color="Green" title="E42">A9</font></span></a>, <a class="txt" href="#E27:43"><span class="lab"><font color="Green" title="E59">A180</font></span></a>, <a class="ref" href="matrix14.html#T10" target="_self" title="MATRIX14:th.10">Th10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:43"><span class="lab"><font color="Green" title="E51">A32</font></span></a>, <a class="txt" href="#E13:43"><span class="lab"><font color="Green" title="E46">A27</font></span></a>, <a class="ref" href="matrix14.html#T35" target="_self" title="MATRIX14:th.35">Th35</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:43"><span class="lab"><font color="Green" title="E52">A33</font></span></a>, <a class="ref" href="polynom1.html#D2" title="POLYNOM1:def.2">POLYNOM1:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:43"><span class="lab"><font color="Green" title="E46">A27</font></span></a>, <a class="txt" href="#E29:43"><span class="lab"><font color="Green" title="E61">A182</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:43"><span class="lab"><font color="Green" title="E55">A70</font></span></a>, <a class="txt" href="#E16:43"><span class="lab"><font color="Green" title="E49">A30</font></span></a>, <a class="ref" href="matrix12.html#T3" title="MATRIX12:th.3">MATRIX12:3</a></span>
<br/>.= 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="txt" href="#E5:43"><span class="lab"><font color="Green" title="E38">A5</font></span></a>, <a class="ref" href="vectsp_1.html#D22" title="VECTSP_1:def.22">VECTSP_1:def 22</a></span>
;<br/>
<a NAME="E32:43"/>
for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">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="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_10"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">i</font>, <font color="Maroon" title="c7">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="c6">i</font>,<font color="Maroon" title="c7">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="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c8">i0</font> = <font color="Maroon" title="c6">i</font>, <font color="Maroon" title="c9">j0</font> = <font color="Maroon" title="c7">j</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <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/>
<span class="kw">assume </span><a NAME="E2:43_10"/><span class="lab"><font color="Green" title="E63">A184</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">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="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></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"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span><br/>

<a NAME="E3:43_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A185</font></span>: 
<font color="Maroon" title="c6">i</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> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:43"><span class="lab"><font color="Green" title="E60">A181</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E4:43_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A186</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E5:43_10"/><span class="lab"><font color="Green" title="E66">A187</font></span>: 
<font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_10"><span class="lab"><font color="Green" title="E64">A185</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E6:43_10"/><span class="lab"><font color="Green" title="E67">A188</font></span>: 
<font color="Maroon" title="c7">j</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> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:43"><span class="lab"><font color="Green" title="E60">A181</font></span></a>, <a class="txt" href="#E2:43_10"><span class="lab"><font color="Green" title="E63">A184</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E7:43_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A189</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E8:43_10"/><span class="lab"><font color="Green" title="E69">A190</font></span>: 
<font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:43_10"><span class="lab"><font color="Green" title="E67">A188</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_10_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="E1:43_10_1"/>
( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">i</font> or 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_10"><span class="lab"><font color="Green" title="E65">A186</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_10_1_1"><span class="kw">suppose </span></a><a NAME="E1:43_10_1_1"/><span class="lab"><font color="Green" title="E70">A191</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">i</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="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_10_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_10_1_1_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="E1:43_10_1_1_1_1"/>
( <font color="Maroon" title="c6">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">j</font> or <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_10_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:43_10_1_1_1_1_1"/><span class="lab"><font color="Green" title="E71">A192</font></span>: 
<font color="Maroon" title="c6">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">j</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="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span><br/><div class="add"><span class="lab"><font color="Green" title="E72">A193</font></span>: <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> = 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c8">i0</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_10"><span class="lab"><font color="Green" title="E65">A186</font></span></a>, <a class="txt" href="#E5:43_10"><span class="lab"><font color="Green" title="E66">A187</font></span></a>, <a class="txt" href="#E7:43_10"><span class="lab"><font color="Green" title="E68">A189</font></span></a>, <a class="txt" href="#E8:43_10"><span class="lab"><font color="Green" title="E69">A190</font></span></a>, <a class="ref" href="matrix14.html#T27" target="_self" title="MATRIX14:th.27">Th27</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:43_10"><span class="lab"><font color="Green" title="E68">A189</font></span></a>, <a class="txt" href="#E8:43_10"><span class="lab"><font color="Green" title="E69">A190</font></span></a>, <a class="txt" href="#E1:43_10_1_1_1_1_1"><span class="lab"><font color="Green" title="E71">A192</font></span></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">Th25</a></span>
;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">i0</font>,<font color="Maroon" title="c9">j0</font>

<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:43"><span class="lab"><font color="Green" title="E58">A115</font></span></a>, <a class="txt" href="#E5:43_10"><span class="lab"><font color="Green" title="E66">A187</font></span></a>, <a class="txt" href="#E7:43_10"><span class="lab"><font color="Green" title="E68">A189</font></span></a>, <a class="txt" href="#E8:43_10"><span class="lab"><font color="Green" title="E69">A190</font></span></a>, <a class="txt" href="#E1:43_10_1_1"><span class="lab"><font color="Green" title="E70">A191</font></span></a>, <a class="txt" href="#E1:43_10_1_1_1_1_1"><span class="lab"><font color="Green" title="E71">A192</font></span></a>, <a class="txt" href="#E2:43_10_1_1_1_1_1"><span class="lab"><font color="Green" title="E72">A193</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_10_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:43_10_1_1_1_1_2"/><span class="lab"><font color="Green" title="E71">A194</font></span>: 
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</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="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span><br/><div class="add"><span class="lab"><font color="Green" title="E72">A195</font></span>: <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> = 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c8">i0</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_10"><span class="lab"><font color="Green" title="E65">A186</font></span></a>, <a class="txt" href="#E5:43_10"><span class="lab"><font color="Green" title="E66">A187</font></span></a>, <a class="txt" href="#E7:43_10"><span class="lab"><font color="Green" title="E68">A189</font></span></a>, <a class="txt" href="#E8:43_10"><span class="lab"><font color="Green" title="E69">A190</font></span></a>, <a class="ref" href="matrix14.html#T27" target="_self" title="MATRIX14:th.27">Th27</a></span>
<br/>.= 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:43_10"><span class="lab"><font color="Green" title="E68">A189</font></span></a>, <a class="txt" href="#E8:43_10"><span class="lab"><font color="Green" title="E69">A190</font></span></a>, <a class="txt" href="#E1:43_10_1_1_1_1_2"><span class="lab"><font color="Green" title="E71">A194</font></span></a>, <a class="ref" href="matrix14.html#T24" target="_self" title="MATRIX14:th.24">Th24</a></span>
;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">i0</font>,<font color="Maroon" title="c9">j0</font>

<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:43"><span class="lab"><font color="Green" title="E56">A71</font></span></a>, <a class="txt" href="#E8:43_10"><span class="lab"><font color="Green" title="E69">A190</font></span></a>, <a class="txt" href="#E1:43_10_1_1"><span class="lab"><font color="Green" title="E70">A191</font></span></a>, <a class="txt" href="#E1:43_10_1_1_1_1_2"><span class="lab"><font color="Green" title="E71">A194</font></span></a>, <a class="txt" href="#E2:43_10_1_1_1_1_2"><span class="lab"><font color="Green" title="E72">A195</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:43_10_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font>
 ; <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="43_10_1_2"><span class="kw">suppose </span></a><a NAME="E1:43_10_1_2"/><span class="lab"><font color="Green" title="E70">A196</font></span>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i</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="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_10_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_10_1_2_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="E1:43_10_1_2_1_1"/>
( <font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j</font> or <font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_10_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:43_10_1_2_1_1_1"/><span class="lab"><font color="Green" title="E71">A197</font></span>: 
<font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j</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="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span><br/><div class="add"><span class="lab"><font color="Green" title="E72">A198</font></span>: <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> = 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c8">i0</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_10"><span class="lab"><font color="Green" title="E65">A186</font></span></a>, <a class="txt" href="#E5:43_10"><span class="lab"><font color="Green" title="E66">A187</font></span></a>, <a class="txt" href="#E7:43_10"><span class="lab"><font color="Green" title="E68">A189</font></span></a>, <a class="txt" href="#E8:43_10"><span class="lab"><font color="Green" title="E69">A190</font></span></a>, <a class="ref" href="matrix14.html#T27" target="_self" title="MATRIX14:th.27">Th27</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:43_10"><span class="lab"><font color="Green" title="E68">A189</font></span></a>, <a class="txt" href="#E8:43_10"><span class="lab"><font color="Green" title="E69">A190</font></span></a>, <a class="txt" href="#E1:43_10_1_2_1_1_1"><span class="lab"><font color="Green" title="E71">A197</font></span></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">Th25</a></span>
;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">i0</font>,<font color="Maroon" title="c9">j0</font>

<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:43"><span class="lab"><font color="Green" title="E54">A35</font></span></a>, <a class="txt" href="#E8:43_10"><span class="lab"><font color="Green" title="E69">A190</font></span></a>, <a class="txt" href="#E1:43_10_1_2"><span class="lab"><font color="Green" title="E70">A196</font></span></a>, <a class="txt" href="#E1:43_10_1_2_1_1_1"><span class="lab"><font color="Green" title="E71">A197</font></span></a>, <a class="txt" href="#E2:43_10_1_2_1_1_1"><span class="lab"><font color="Green" title="E72">A198</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_10_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:43_10_1_2_1_1_2"/><span class="lab"><font color="Green" title="E71">A199</font></span>: 
<font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">j</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="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span><br/><div class="add"><a NAME="E2:43_10_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A200</font></span>: 
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:43_10"><span class="lab"><font color="Green" title="E68">A189</font></span></a>, <a class="txt" href="#E1:43_10_1_2"><span class="lab"><font color="Green" title="E70">A196</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> = 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c8">i0</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_10"><span class="lab"><font color="Green" title="E65">A186</font></span></a>, <a class="txt" href="#E5:43_10"><span class="lab"><font color="Green" title="E66">A187</font></span></a>, <a class="txt" href="#E7:43_10"><span class="lab"><font color="Green" title="E68">A189</font></span></a>, <a class="txt" href="#E8:43_10"><span class="lab"><font color="Green" title="E69">A190</font></span></a>, <a class="ref" href="matrix14.html#T27" target="_self" title="MATRIX14:th.27">Th27</a></span>
<br/>.= 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:43_10"><span class="lab"><font color="Green" title="E68">A189</font></span></a>, <a class="txt" href="#E8:43_10"><span class="lab"><font color="Green" title="E69">A190</font></span></a>, <a class="txt" href="#E2:43_10_1_2_1_1_2"><span class="lab"><font color="Green" title="E72">A200</font></span></a>, <a class="ref" href="matrix14.html#T24" target="_self" title="MATRIX14:th.24">Th24</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:43_10_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:43"><span class="lab"><font color="Green" title="E62">A183</font></span></a>, <a class="txt" href="#E7:43_10"><span class="lab"><font color="Green" title="E68">A189</font></span></a>, <a class="txt" href="#E1:43_10_1_2"><span class="lab"><font color="Green" title="E70">A196</font></span></a>, <a class="txt" href="#E1:43_10_1_2_1_1_2"><span class="lab"><font color="Green" title="E71">A199</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</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><span class="kw">hence </span><a NAME="E3:43_10_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font>
 ; <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 NAME="E33:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A201</font></span>: 
<font color="Maroon" title="c5">Q</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T28" title="MATRIX_1:th.28">MATRIX_1:28</a></span>;<br/>
<a NAME="E34:43"/><span class="lab"><font color="Green" title="E64">A202</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></span>;<br/>
<a NAME="E35:43"/><span class="lab"><font color="Green" title="E65">A203</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">Th23</a></span>;<br/>
<span class="lab"><font color="Green" title="E66">A204</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixr1.html#T16" title="MATRIXR1:th.16">MATRIXR1:16</a></span>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">Th23</a></span>
;<br/>
<a NAME="E37:43"/><span class="lab"><font color="Green" title="E67">A205</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> <font color="Maroon" title="c1">n</font> holds <br/><span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span> <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> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> <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="43_12"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">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="c6">k</font> &amp; <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> implies <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:43_12"/><span class="lab"><font color="Green" title="E68">A206</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">k</font>
 <span class="kw">and </span><br/><a NAME="E2:43_12"/><span class="lab"><font color="Green" title="E69">A207</font></span>: 
<font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</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"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font></span><br/>

<a NAME="E3:43_12"/><span class="lab"><font color="Green" title="E70">A208</font></span>: 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:43"><span class="lab"><font color="Green" title="E65">A203</font></span></a>, <a class="txt" href="#E1:43_12"><span class="lab"><font color="Green" title="E68">A206</font></span></a>, <a class="txt" href="#E2:43_12"><span class="lab"><font color="Green" title="E69">A207</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/>
<a NAME="E4:43_12"/><span class="lab"><font color="Green" title="E71">A209</font></span>: 
<font color="Maroon" title="c6">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:43"><span class="lab"><font color="Green" title="E66">A204</font></span></a>, <a class="txt" href="#E1:43_12"><span class="lab"><font color="Green" title="E68">A206</font></span></a>, <a class="txt" href="#E2:43_12"><span class="lab"><font color="Green" title="E69">A207</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E72">A210</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="43_12_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:43_12_1"/><span class="lab"><font color="Green" title="E73">A211</font></span>: 
<font color="Maroon" title="c6">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <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="c3">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span><br/><span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> = 
<font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_12"><span class="lab"><font color="Green" title="E70">A208</font></span></a>, <a class="txt" href="#E4:43_12"><span class="lab"><font color="Green" title="E71">A209</font></span></a>, <a class="ref" href="fvsum_1.html#T62" title="FVSUM_1:th.62">FVSUM_1:62</a></span>
<br/>.= 
<font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_12"><span class="lab"><font color="Green" title="E68">A206</font></span></a>, <a class="txt" href="#E2:43_12"><span class="lab"><font color="Green" title="E69">A207</font></span></a>, <a class="txt" href="#E3:43_12"><span class="lab"><font color="Green" title="E70">A208</font></span></a>, <a class="txt" href="#E1:43_12_1"><span class="lab"><font color="Green" title="E73">A211</font></span></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">Th25</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</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>
<a NAME="E6:43_12"/><span class="lab"><font color="Green" title="E73">A212</font></span>: 
<font color="Maroon" title="c6">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="E7:43_12"/>
<font color="Maroon" title="c6">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> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_12"><span class="lab"><font color="Green" title="E68">A206</font></span></a>, <a class="txt" href="#E2:43_12"><span class="lab"><font color="Green" title="E69">A207</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E8:43_12"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A213</font></span>: 
<font color="Maroon" title="c6">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">Q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:43"><span class="lab"><font color="Green" title="E64">A202</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E75">A214</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="43_12_2"><span class="kw">now </span></a><div class="add"><a NAME="E1:43_12_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">k</font>,1</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> <font color="Maroon" title="c5">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:43"><span class="lab"><font color="Green" title="E41">A8</font></span></a>, <a class="txt" href="#E1:43_12"><span class="lab"><font color="Green" title="E68">A206</font></span></a>, <a class="txt" href="#E2:43_12"><span class="lab"><font color="Green" title="E69">A207</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E2:43_12_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A215</font></span>: 
 ex <font color="Olive" title="b1">p</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font> st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> &amp; <font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">k</font>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</a></span>;<br/><span class="kw">assume </span><a NAME="E3:43_12_2"/><span class="lab"><font color="Green" title="E77">A216</font></span>: 
<font color="Maroon" title="c6">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <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"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span><br/><a NAME="E4:43_12_2"/><span class="kw">then </span>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_12"><span class="lab"><font color="Green" title="E68">A206</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">then </span><font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">k</font>,1 = 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:43"><span class="lab"><font color="Green" title="E40">A7</font></span></a>, <a class="txt" href="#E2:43_12"><span class="lab"><font color="Green" title="E69">A207</font></span></a>, <a class="txt" href="#E6:43_12"><span class="lab"><font color="Green" title="E73">A212</font></span></a>, <a class="txt" href="#E2:43_12_2"><span class="lab"><font color="Green" title="E76">A215</font></span></a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:43"><span class="lab"><font color="Green" title="E41">A8</font></span></a>, <a class="txt" href="#E3:43_12_2"><span class="lab"><font color="Green" title="E77">A216</font></span></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">Th25</a></span>
;<br/><span class="kw">hence </span><a NAME="E6:43_12_2"/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:43_12"><span class="lab"><font color="Green" title="E74">A213</font></span></a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</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="lab"><font color="Green" title="E76">A217</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="43_12_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:43_12_3"/><span class="lab"><font color="Green" title="E77">A218</font></span>: 
<font color="Maroon" title="c6">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"> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font></span><br/><span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> = 
<font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_12"><span class="lab"><font color="Green" title="E70">A208</font></span></a>, <a class="txt" href="#E4:43_12"><span class="lab"><font color="Green" title="E71">A209</font></span></a>, <a class="ref" href="fvsum_1.html#T62" title="FVSUM_1:th.62">FVSUM_1:62</a></span>
<br/>.= 
<font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_12"><span class="lab"><font color="Green" title="E69">A207</font></span></a>, <a class="txt" href="#E3:43_12"><span class="lab"><font color="Green" title="E70">A208</font></span></a>, <a class="txt" href="#E1:43_12_3"><span class="lab"><font color="Green" title="E77">A218</font></span></a>, <a class="ref" href="matrix14.html#T24" target="_self" title="MATRIX14:th.24">Th24</a></span>
<br/>.= 
<font color="Maroon" title="c3">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D16" title="VECTSP_1:def.16">VECTSP_1:def 16</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>
<a NAME="E11:43_12"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_12"><span class="lab"><font color="Green" title="E68">A206</font></span></a>, <a class="txt" href="#E2:43_12"><span class="lab"><font color="Green" title="E69">A207</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E12:43_12"/><span class="kw">then </span>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">Q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:43"><span class="lab"><font color="Green" title="E64">A202</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:43_12"/>
<span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43"><span class="lab"><font color="Green" title="E38">A5</font></span></a>, <a class="txt" href="#E5:43_12"><span class="lab"><font color="Green" title="E72">A210</font></span></a>, <a class="txt" href="#E10:43_12"><span class="lab"><font color="Green" title="E76">A217</font></span></a>, <a class="txt" href="#E9:43_12"><span class="lab"><font color="Green" title="E75">A214</font></span></a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</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>
<a NAME="E38:43"/><span class="lab"><font color="Green" title="E68">A219</font></span>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E39:43"/><span class="lab"><font color="Green" title="E69">A220</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></span>;<br/>
<a NAME="E40:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A221</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">P</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:43"><span class="lab"><font color="Green" title="E68">A219</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E41:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A222</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">P</font>
 
<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="E42:43"/><span class="lab"><font color="Green" title="E72">A223</font></span>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></span>;<br/>
<a NAME="E43:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A224</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>;<br/>
<a NAME="E44:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A225</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:43"><span class="lab"><font color="Green" title="E69">A220</font></span></a>, <a class="txt" href="#E40:43"><span class="lab"><font color="Green" title="E70">A221</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E45:43"/><span class="lab"><font color="Green" title="E75">A226</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c4">P</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:43"><span class="lab"><font color="Green" title="E72">A223</font></span></a>, <a class="txt" href="#E38:43"><span class="lab"><font color="Green" title="E68">A219</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E46:43"/><span class="lab"><font color="Green" title="E76">A227</font></span>: 
for <font color="Olive" title="b1">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_13"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">j</font> &amp; <font color="Maroon" title="c6">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:43_13"/><span class="lab"><font color="Green" title="E77">A228</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">j</font>
 <span class="kw">and </span><br/><a NAME="E2:43_13"/><span class="lab"><font color="Green" title="E78">A229</font></span>: 
<font color="Maroon" title="c6">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</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="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span><br/>

<span class="lab"><font color="Green" title="E79">A230</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix14.html#T6" target="_self" title="MATRIX14:th.6">Th6</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixr1.html#T16" title="MATRIXR1:th.16">MATRIXR1:16</a></span>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">Th23</a></span>
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">p</font> =  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">j</font>, <font color="Maroon" title="c8">q</font> = <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font> ;<br/>
<a NAME="E5:43_13"/><span class="lab"><font color="Green" title="E80">A231</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">Th23</a></span>;<br/>
<span class="lab"><font color="Green" title="E81">A232</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixr1.html#T16" title="MATRIXR1:th.16">MATRIXR1:16</a></span>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">Th23</a></span>
;<br/>
<a NAME="E7:43_13"/><span class="lab"><font color="Green" title="E82">A233</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> <font color="Maroon" title="c1">n</font> holds <br/><font color="Maroon" title="c7">p</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="c8">q</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="43_13_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">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="c9">k</font> &amp; <font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> implies <font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:43_13_3"/><span class="lab"><font color="Green" title="E83">A234</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">k</font>
 <span class="kw">and </span><br/><a NAME="E2:43_13_3"/><span class="lab"><font color="Green" title="E84">A235</font></span>: 
<font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</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="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font></span><br/>

<a NAME="E3:43_13_3"/><span class="lab"><font color="Green" title="E85">A236</font></span>: 
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">Q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:43"><span class="lab"><font color="Green" title="E64">A202</font></span></a>, <a class="txt" href="#E1:43_13_3"><span class="lab"><font color="Green" title="E83">A234</font></span></a>, <a class="txt" href="#E2:43_13_3"><span class="lab"><font color="Green" title="E84">A235</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E4:43_13_3"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">Th23</a></span>;<br/>
<a NAME="E5:43_13_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A237</font></span>: 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_13_3"><span class="lab"><font color="Green" title="E83">A234</font></span></a>, <a class="txt" href="#E2:43_13_3"><span class="lab"><font color="Green" title="E84">A235</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/>
<a NAME="E6:43_13_3"/><span class="lab"><font color="Green" title="E87">A238</font></span>: 
( <font color="Maroon" title="c9">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:43"><span class="lab"><font color="Green" title="E65">A203</font></span></a>, <a class="txt" href="#E6:43_13"><span class="lab"><font color="Green" title="E81">A232</font></span></a>, <a class="txt" href="#E1:43_13_3"><span class="lab"><font color="Green" title="E83">A234</font></span></a>, <a class="txt" href="#E2:43_13_3"><span class="lab"><font color="Green" title="E84">A235</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/>
<span class="lab"><font color="Green" title="E88">A239</font></span>: <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix14.html#T6" target="_self" title="MATRIX14:th.6">Th6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:43_13_3"><span class="lab"><font color="Green" title="E87">A238</font></span></a>, <a class="ref" href="fvsum_1.html#T62" title="FVSUM_1:th.62">FVSUM_1:62</a></span>
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_13_3_2"><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="E1:43_13_3_2"/>
( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k</font> or 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">k</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_13_3"><span class="lab"><font color="Green" title="E83">A234</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_13_3_2_1"><span class="kw">suppose </span></a><a NAME="E1:43_13_3_2_1"/><span class="lab"><font color="Green" title="E89">A240</font></span>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k</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="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font></span><br/><div class="add"><span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font> = 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43_13"><span class="lab"><font color="Green" title="E80">A231</font></span></a>, <a class="txt" href="#E1:43_13_3"><span class="lab"><font color="Green" title="E83">A234</font></span></a>, <a class="txt" href="#E2:43_13_3"><span class="lab"><font color="Green" title="E84">A235</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_13"><span class="lab"><font color="Green" title="E77">A228</font></span></a>, <a class="txt" href="#E2:43_13_3"><span class="lab"><font color="Green" title="E84">A235</font></span></a>, <a class="txt" href="#E1:43_13_3_2_1"><span class="lab"><font color="Green" title="E89">A240</font></span></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">Th25</a></span>
;<br/><a NAME="E3:43_13_3_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p4">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_13"><span class="lab"><font color="Green" title="E79">A230</font></span></a>, <a class="txt" href="#E5:43_13"><span class="lab"><font color="Green" title="E80">A231</font></span></a>, <a class="txt" href="#E1:43_13_3"><span class="lab"><font color="Green" title="E83">A234</font></span></a>, <a class="txt" href="#E2:43_13_3"><span class="lab"><font color="Green" title="E84">A235</font></span></a>, <a class="ref" href="matrix14.html#T5" target="_self" title="MATRIX14:th.5">Th5</a></span>;<br/><a NAME="E4:43_13_3_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A241</font></span>: 
<font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></span>;<br/><span class="lab"><font color="Green" title="E91">A242</font></span>: <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> = 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">K</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_13_3"><span class="lab"><font color="Green" title="E84">A235</font></span></a>, <a class="txt" href="#E7:43_13_3"><span class="lab"><font color="Green" title="E88">A239</font></span></a>, <a class="txt" href="#E5:43_13_3"><span class="lab"><font color="Green" title="E86">A237</font></span></a>, <a class="txt" href="#E1:43_13_3_2_1"><span class="lab"><font color="Green" title="E89">A240</font></span></a>, <a class="ref" href="matrix14.html#T24" target="_self" title="MATRIX14:th.24">Th24</a></span>
<br/>.= 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D5" title="GROUP_1:def.5">GROUP_1:def 5</a></span>
;<br/><font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> = 
<font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_13_3"><span class="lab"><font color="Green" title="E85">A236</font></span></a>, <a class="txt" href="#E1:43_13_3_2_1"><span class="lab"><font color="Green" title="E89">A240</font></span></a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></span>
<br/>.= 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:43"><span class="lab"><font color="Green" title="E39">A6</font></span></a>, <a class="txt" href="#E1:43_13"><span class="lab"><font color="Green" title="E77">A228</font></span></a>, <a class="txt" href="#E2:43_13"><span class="lab"><font color="Green" title="E78">A229</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E7:43_13_3_2_1"/>
<font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_13"><span class="lab"><font color="Green" title="E79">A230</font></span></a>, <a class="txt" href="#E1:43_13_3"><span class="lab"><font color="Green" title="E83">A234</font></span></a>, <a class="txt" href="#E2:43_13_3"><span class="lab"><font color="Green" title="E84">A235</font></span></a>, <a class="txt" href="#E5:43_13_3_2_1"><span class="lab"><font color="Green" title="E91">A242</font></span></a>, <a class="txt" href="#E4:43_13_3_2_1"><span class="lab"><font color="Green" title="E90">A241</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</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="43_13_3_2_2"><span class="kw">suppose </span></a><a NAME="E1:43_13_3_2_2"/><span class="lab"><font color="Green" title="E89">A243</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">k</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="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font></span><br/><div class="add"><a NAME="E2:43_13_3_2_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">k</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> <font color="Maroon" title="c5">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_13"><span class="lab"><font color="Green" title="E77">A228</font></span></a>, <a class="txt" href="#E2:43_13"><span class="lab"><font color="Green" title="E78">A229</font></span></a>, <a class="txt" href="#E1:43_13_3"><span class="lab"><font color="Green" title="E83">A234</font></span></a>, <a class="txt" href="#E2:43_13_3"><span class="lab"><font color="Green" title="E84">A235</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">p2</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font><span class="kw"> such that </span><br/><a NAME="E4:43_13_3_2_2"/><span class="lab"><font color="Green" title="E90">A244</font></span>: 
<font color="Maroon" title="c10">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font>
 <span class="kw">and </span><br/><a NAME="E5:43_13_3_2_2"/><span class="lab"><font color="Green" title="E91">A245</font></span>: 
<font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</a></span>;<br/><a NAME="E6:43_13_3_2_2"/><span class="lab"><font color="Green" title="E92">A246</font></span>: 
<font color="Maroon" title="c9">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/><span class="lab"><font color="Green" title="E93">A247</font></span>: <font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> = 
<font color="Maroon" title="c10">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_13_3"><span class="lab"><font color="Green" title="E85">A236</font></span></a>, <a class="txt" href="#E5:43_13_3_2_2"><span class="lab"><font color="Green" title="E91">A245</font></span></a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c9">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:43"><span class="lab"><font color="Green" title="E40">A7</font></span></a>, <a class="txt" href="#E2:43_13_3"><span class="lab"><font color="Green" title="E84">A235</font></span></a>, <a class="txt" href="#E1:43_13_3_2_2"><span class="lab"><font color="Green" title="E89">A243</font></span></a>, <a class="txt" href="#E4:43_13_3_2_2"><span class="lab"><font color="Green" title="E90">A244</font></span></a>, <a class="txt" href="#E6:43_13_3_2_2"><span class="lab"><font color="Green" title="E92">A246</font></span></a></span>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_13_3_2_2_2"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_13_3_2_2_2_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="E1:43_13_3_2_2_2_1"/>
( <font color="Maroon" title="c9">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">j</font> or <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_13_3_2_2_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:43_13_3_2_2_2_1_1"/><span class="lab"><font color="Green" title="E94">A248</font></span>: 
<font color="Maroon" title="c9">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">j</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="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font></span><br/><div class="add"><span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font> = 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_13"><span class="lab"><font color="Green" title="E79">A230</font></span></a>, <a class="txt" href="#E1:43_13_3"><span class="lab"><font color="Green" title="E83">A234</font></span></a>, <a class="txt" href="#E2:43_13_3"><span class="lab"><font color="Green" title="E84">A235</font></span></a>, <a class="txt" href="#E7:43_13_3"><span class="lab"><font color="Green" title="E88">A239</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_13_3"><span class="lab"><font color="Green" title="E84">A235</font></span></a>, <a class="txt" href="#E5:43_13_3"><span class="lab"><font color="Green" title="E86">A237</font></span></a>, <a class="txt" href="#E1:43_13_3_2_2"><span class="lab"><font color="Green" title="E89">A243</font></span></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">Th25</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></span>
;<br/><span class="kw">then </span><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> = 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</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"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_13"><span class="lab"><font color="Green" title="E79">A230</font></span></a>, <a class="txt" href="#E5:43_13"><span class="lab"><font color="Green" title="E80">A231</font></span></a>, <a class="txt" href="#E1:43_13_3"><span class="lab"><font color="Green" title="E83">A234</font></span></a>, <a class="txt" href="#E2:43_13_3"><span class="lab"><font color="Green" title="E84">A235</font></span></a>, <a class="ref" href="matrix14.html#T5" target="_self" title="MATRIX14:th.5">Th5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43_13"><span class="lab"><font color="Green" title="E80">A231</font></span></a>, <a class="txt" href="#E1:43_13_3"><span class="lab"><font color="Green" title="E83">A234</font></span></a>, <a class="txt" href="#E2:43_13_3"><span class="lab"><font color="Green" title="E84">A235</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_13_3"><span class="lab"><font color="Green" title="E83">A234</font></span></a>, <a class="txt" href="#E2:43_13_3"><span class="lab"><font color="Green" title="E84">A235</font></span></a>, <a class="txt" href="#E1:43_13_3_2_2_2_1_1"><span class="lab"><font color="Green" title="E94">A248</font></span></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">Th25</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:43_13_3_2_2_2_1_1"/>
<font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_13"><span class="lab"><font color="Green" title="E77">A228</font></span></a>, <a class="txt" href="#E2:43_13"><span class="lab"><font color="Green" title="E78">A229</font></span></a>, <a class="txt" href="#E7:43_13_3_2_2"><span class="lab"><font color="Green" title="E93">A247</font></span></a>, <a class="txt" href="#E1:43_13_3_2_2_2_1_1"><span class="lab"><font color="Green" title="E94">A248</font></span></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">Th25</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="43_13_3_2_2_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:43_13_3_2_2_2_1_2"/><span class="lab"><font color="Green" title="E94">A249</font></span>: 
<font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">j</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="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font></span><br/><div class="add"><span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font> = 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_13"><span class="lab"><font color="Green" title="E79">A230</font></span></a>, <a class="txt" href="#E1:43_13_3"><span class="lab"><font color="Green" title="E83">A234</font></span></a>, <a class="txt" href="#E2:43_13_3"><span class="lab"><font color="Green" title="E84">A235</font></span></a>, <a class="txt" href="#E7:43_13_3"><span class="lab"><font color="Green" title="E88">A239</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_13_3"><span class="lab"><font color="Green" title="E84">A235</font></span></a>, <a class="txt" href="#E5:43_13_3"><span class="lab"><font color="Green" title="E86">A237</font></span></a>, <a class="txt" href="#E1:43_13_3_2_2"><span class="lab"><font color="Green" title="E89">A243</font></span></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">Th25</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></span>
;<br/><span class="kw">then </span><font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> = 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</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"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_13"><span class="lab"><font color="Green" title="E79">A230</font></span></a>, <a class="txt" href="#E5:43_13"><span class="lab"><font color="Green" title="E80">A231</font></span></a>, <a class="txt" href="#E1:43_13_3"><span class="lab"><font color="Green" title="E83">A234</font></span></a>, <a class="txt" href="#E2:43_13_3"><span class="lab"><font color="Green" title="E84">A235</font></span></a>, <a class="ref" href="matrix14.html#T5" target="_self" title="MATRIX14:th.5">Th5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43_13"><span class="lab"><font color="Green" title="E80">A231</font></span></a>, <a class="txt" href="#E1:43_13_3"><span class="lab"><font color="Green" title="E83">A234</font></span></a>, <a class="txt" href="#E2:43_13_3"><span class="lab"><font color="Green" title="E84">A235</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>
<br/>.= 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_13_3"><span class="lab"><font color="Green" title="E83">A234</font></span></a>, <a class="txt" href="#E2:43_13_3"><span class="lab"><font color="Green" title="E84">A235</font></span></a>, <a class="txt" href="#E1:43_13_3_2_2_2_1_2"><span class="lab"><font color="Green" title="E94">A249</font></span></a>, <a class="ref" href="matrix14.html#T24" target="_self" title="MATRIX14:th.24">Th24</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:43_13_3_2_2_2_1_2"/>
<font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_13_3"><span class="lab"><font color="Green" title="E83">A234</font></span></a>, <a class="txt" href="#E2:43_13_3"><span class="lab"><font color="Green" title="E84">A235</font></span></a>, <a class="txt" href="#E7:43_13_3_2_2"><span class="lab"><font color="Green" title="E93">A247</font></span></a>, <a class="txt" href="#E1:43_13_3_2_2_2_1_2"><span class="lab"><font color="Green" title="E94">A249</font></span></a>, <a class="ref" href="matrix14.html#T24" target="_self" title="MATRIX14:th.24">Th24</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><span class="kw">hence </span><a NAME="E9:43_13_3_2_2"/>
<font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">k</font>
 ; <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 NAME="E8:43_13"/><span class="lab"><font color="Green" title="E83">A250</font></span>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></span>;<br/>
<a NAME="E9:43_13"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A251</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E85">A252</font></span>: <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:43"><span class="lab"><font color="Green" title="E68">A219</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>
<br/>.= 
<font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E36">A3</font></span></a>, <a class="txt" href="#E45:43"><span class="lab"><font color="Green" title="E75">A226</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>
;<br/>
<a NAME="E11:43_13"/><span class="lab"><font color="Green" title="E86">A253</font></span>: 
<font color="Maroon" title="c6">j</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> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_13"><span class="lab"><font color="Green" title="E77">A228</font></span></a>, <a class="txt" href="#E2:43_13"><span class="lab"><font color="Green" title="E78">A229</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<span class="lab"><font color="Green" title="E87">A254</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">j</font></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">Q</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></span>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></span>
;<br/>
<a NAME="E13:43_13"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p4">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_13"><span class="lab"><font color="Green" title="E79">A230</font></span></a>, <a class="txt" href="#E5:43_13"><span class="lab"><font color="Green" title="E80">A231</font></span></a>, <a class="ref" href="matrix14.html#T2" target="_self" title="MATRIX14:th.2">Th2</a></span>;<br/>
<a NAME="E14:43_13"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A255</font></span>: 
 <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:43_13"><span class="lab"><font color="Green" title="E87">A254</font></span></a>, <a class="txt" href="#E7:43_13"><span class="lab"><font color="Green" title="E82">A233</font></span></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></span>;<br/>
<span class="lab"><font color="Green" title="E89">A256</font></span>: <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">j</font> = 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_13"><span class="lab"><font color="Green" title="E77">A228</font></span></a>, <a class="txt" href="#E2:43_13"><span class="lab"><font color="Green" title="E78">A229</font></span></a>, <a class="txt" href="#E9:43_13"><span class="lab"><font color="Green" title="E84">A251</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>
<br/>.= 
<font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:43_13"><span class="lab"><font color="Green" title="E86">A253</font></span></a>, <a class="txt" href="#E8:43_13"><span class="lab"><font color="Green" title="E83">A250</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>
;<br/>
<a NAME="E16:43_13"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,<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="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:43"><span class="lab"><font color="Green" title="E68">A219</font></span></a>, <a class="txt" href="#E1:43_13"><span class="lab"><font color="Green" title="E77">A228</font></span></a>, <a class="txt" href="#E2:43_13"><span class="lab"><font color="Green" title="E78">A229</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font> = 
<span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c6">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:43"><span class="lab"><font color="Green" title="E64">A202</font></span></a>, <a class="txt" href="#E8:43_13"><span class="lab"><font color="Green" title="E83">A250</font></span></a>, <a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a></span>
<br/>.= 
<span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p4">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:43"><span class="lab"><font color="Green" title="E73">A224</font></span></a>, <a class="txt" href="#E3:43_13"><span class="lab"><font color="Green" title="E79">A230</font></span></a>, <a class="txt" href="#E5:43_13"><span class="lab"><font color="Green" title="E80">A231</font></span></a>, <a class="txt" href="#E14:43_13"><span class="lab"><font color="Green" title="E88">A255</font></span></a>, <a class="ref" href="matrix14.html#T11" target="_self" title="MATRIX14:th.11">Th11</a></span>
<br/>.= 
<span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix14.html#T6" target="_self" title="MATRIX14:th.6">Th6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,1</span>)</span>,<span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:43"><span class="lab"><font color="Green" title="E73">A224</font></span></a>, <a class="txt" href="#E35:43"><span class="lab"><font color="Green" title="E65">A203</font></span></a>, <a class="ref" href="matrix14.html#T10" target="_self" title="MATRIX14:th.10">Th10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:43"><span class="lab"><font color="Green" title="E73">A224</font></span></a>, <a class="txt" href="#E38:43"><span class="lab"><font color="Green" title="E68">A219</font></span></a>, <a class="txt" href="#E10:43_13"><span class="lab"><font color="Green" title="E85">A252</font></span></a>, <a class="ref" href="matrix14.html#T35" target="_self" title="MATRIX14:th.35">Th35</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T41" title="VECTSP_1:th.41">VECTSP_1:41</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="ref" href="vectsp_1.html#D22" title="VECTSP_1:def.22">VECTSP_1:def 22</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span></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"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:43"><span class="lab"><font color="Green" title="E73">A224</font></span></a>, <a class="txt" href="#E1:43_13"><span class="lab"><font color="Green" title="E77">A228</font></span></a>, <a class="txt" href="#E2:43_13"><span class="lab"><font color="Green" title="E78">A229</font></span></a>, <a class="ref" href="matrix14.html#T35" target="_self" title="MATRIX14:th.35">Th35</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:43_13"><span class="lab"><font color="Green" title="E89">A256</font></span></a>, <a class="ref" href="vectsp_1.html#D16" title="VECTSP_1:def.16">VECTSP_1:def 16</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T16" title="RLVECT_1:th.16">RLVECT_1:16</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E18:43_13"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E47:43"/><span class="lab"><font color="Green" title="E77">A257</font></span>: 
 <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c5">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<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>,<span class="p2">(<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></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></span>;<br/>
<a NAME="E48:43"/><span class="lab"><font color="Green" title="E78">A258</font></span>: 
for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">j</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_14"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">i</font>, <font color="Maroon" title="c7">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">i</font> &amp; <font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:43_14"/><span class="lab"><font color="Green" title="E79">A259</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">i</font>
 <span class="kw">and </span><br/><a NAME="E2:43_14"/><span class="lab"><font color="Green" title="E80">A260</font></span>: 
<font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">and </span><br/><a NAME="E3:43_14"/><span class="lab"><font color="Green" title="E81">A261</font></span>: 
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</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="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font></span><br/>

<span class="lab"><font color="Green" title="E82">A262</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix14.html#T6" target="_self" title="MATRIX14:th.6">Th6</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixr1.html#T16" title="MATRIXR1:th.16">MATRIXR1:16</a></span>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">Th23</a></span>
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">p</font> =  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c7">j</font>, <font color="Maroon" title="c9">q</font> = <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font> ;<br/>
<a NAME="E6:43_14"/><span class="lab"><font color="Green" title="E83">A263</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">Th23</a></span>;<br/>
<a NAME="E7:43_14"/><span class="lab"><font color="Green" title="E84">A264</font></span>: 
<font color="Maroon" title="c7">j</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> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_14"><span class="lab"><font color="Green" title="E79">A259</font></span></a>, <a class="txt" href="#E2:43_14"><span class="lab"><font color="Green" title="E80">A260</font></span></a>, <a class="txt" href="#E3:43_14"><span class="lab"><font color="Green" title="E81">A261</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E8:43_14"/><span class="lab"><font color="Green" title="E85">A265</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> <font color="Maroon" title="c1">n</font> holds <br/><font color="Maroon" title="c8">p</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="c9">q</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="43_14_2"><span class="kw">proof </span></a><div class="add">

<span class="lab"><font color="Green" title="E86">A266</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixr1.html#T16" title="MATRIXR1:th.16">MATRIXR1:16</a></span>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">Th23</a></span>
;<br/>
<span class="kw">let </span><font color="Maroon" title="c10">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="c10">k</font> &amp; <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> implies <font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:43_14_2"/><span class="lab"><font color="Green" title="E87">A267</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font>
 <span class="kw">and </span><br/><a NAME="E3:43_14_2"/><span class="lab"><font color="Green" title="E88">A268</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</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="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/>

<a NAME="E4:43_14_2"/><span class="lab"><font color="Green" title="E89">A269</font></span>: 
<font color="Maroon" title="c10">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> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_14_2"><span class="lab"><font color="Green" title="E87">A267</font></span></a>, <a class="txt" href="#E3:43_14_2"><span class="lab"><font color="Green" title="E88">A268</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixr1.html#T16" title="MATRIXR1:th.16">MATRIXR1:16</a></span>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">Th23</a></span>
;<br/>
<a NAME="E6:43_14_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A270</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_14_2"><span class="lab"><font color="Green" title="E87">A267</font></span></a>, <a class="txt" href="#E3:43_14_2"><span class="lab"><font color="Green" title="E88">A268</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E7:43_14_2"/><span class="lab"><font color="Green" title="E91">A271</font></span>: 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:43"><span class="lab"><font color="Green" title="E65">A203</font></span></a>, <a class="txt" href="#E2:43_14_2"><span class="lab"><font color="Green" title="E87">A267</font></span></a>, <a class="txt" href="#E3:43_14_2"><span class="lab"><font color="Green" title="E88">A268</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/>
<a NAME="E8:43_14_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">Th23</a></span>;<br/>
<a NAME="E9:43_14_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A272</font></span>: 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_14_2"><span class="lab"><font color="Green" title="E87">A267</font></span></a>, <a class="txt" href="#E3:43_14_2"><span class="lab"><font color="Green" title="E88">A268</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/>
<a NAME="E10:43_14_2"/>
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">Q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:43"><span class="lab"><font color="Green" title="E64">A202</font></span></a>, <a class="txt" href="#E2:43_14_2"><span class="lab"><font color="Green" title="E87">A267</font></span></a>, <a class="txt" href="#E3:43_14_2"><span class="lab"><font color="Green" title="E88">A268</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E11:43_14_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A273</font></span>: 
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c7">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></span>;<br/>
<span class="lab"><font color="Green" title="E94">A274</font></span>: <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix14.html#T6" target="_self" title="MATRIX14:th.6">Th6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:43_14_2"><span class="lab"><font color="Green" title="E90">A270</font></span></a>, <a class="txt" href="#E7:43_14_2"><span class="lab"><font color="Green" title="E91">A271</font></span></a>, <a class="ref" href="fvsum_1.html#T62" title="FVSUM_1:th.62">FVSUM_1:62</a></span>
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_14_2_4"><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="E1:43_14_2_4"/>
( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</font> or 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">k</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_14_2"><span class="lab"><font color="Green" title="E87">A267</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_14_2_4_1"><span class="kw">suppose </span></a><a NAME="E1:43_14_2_4_1"/><span class="lab"><font color="Green" title="E95">A275</font></span>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</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="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/><div class="add"><a NAME="E2:43_14_2_4_1"/>
<font color="Maroon" title="c10">k</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> <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_14_2"><span class="lab"><font color="Green" title="E88">A268</font></span></a>, <a class="txt" href="#E1:43_14_2"><span class="lab"><font color="Green" title="E86">A266</font></span></a>, <a class="ref" href="matrix14.html#T6" target="_self" title="MATRIX14:th.6">Th6</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E96">A276</font></span>: <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_14_2"><span class="lab"><font color="Green" title="E87">A267</font></span></a>, <a class="txt" href="#E12:43_14_2"><span class="lab"><font color="Green" title="E94">A274</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:43"><span class="lab"><font color="Green" title="E68">A219</font></span></a>, <a class="txt" href="#E9:43_14_2"><span class="lab"><font color="Green" title="E92">A272</font></span></a>, <a class="txt" href="#E1:43_14_2_4_1"><span class="lab"><font color="Green" title="E95">A275</font></span></a>, <a class="ref" href="matrix14.html#T24" target="_self" title="MATRIX14:th.24">Th24</a></span>
;<br/><span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:43_14"><span class="lab"><font color="Green" title="E83">A263</font></span></a>, <a class="txt" href="#E2:43_14_2"><span class="lab"><font color="Green" title="E87">A267</font></span></a>, <a class="txt" href="#E3:43_14_2"><span class="lab"><font color="Green" title="E88">A268</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_14"><span class="lab"><font color="Green" title="E79">A259</font></span></a>, <a class="txt" href="#E3:43_14"><span class="lab"><font color="Green" title="E81">A261</font></span></a>, <a class="txt" href="#E3:43_14_2"><span class="lab"><font color="Green" title="E88">A268</font></span></a>, <a class="txt" href="#E1:43_14_2_4_1"><span class="lab"><font color="Green" title="E95">A275</font></span></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">Th25</a></span>
;<br/><span class="kw">then </span><font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_14"><span class="lab"><font color="Green" title="E82">A262</font></span></a>, <a class="txt" href="#E6:43_14"><span class="lab"><font color="Green" title="E83">A263</font></span></a>, <a class="txt" href="#E2:43_14_2"><span class="lab"><font color="Green" title="E87">A267</font></span></a>, <a class="txt" href="#E3:43_14_2"><span class="lab"><font color="Green" title="E88">A268</font></span></a>, <a class="txt" href="#E3:43_14_2_4_1"><span class="lab"><font color="Green" title="E96">A276</font></span></a>, <a class="ref" href="matrix14.html#T5" target="_self" title="MATRIX14:th.5">Th5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></span>
<br/>.= 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D16" title="VECTSP_1:def.16">VECTSP_1:def 16</a></span>
;<br/><span class="kw">hence </span><a NAME="E6:43_14_2_4_1"/>
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:43"><span class="lab"><font color="Green" title="E39">A6</font></span></a>, <a class="txt" href="#E1:43_14"><span class="lab"><font color="Green" title="E79">A259</font></span></a>, <a class="txt" href="#E2:43_14"><span class="lab"><font color="Green" title="E80">A260</font></span></a>, <a class="txt" href="#E3:43_14"><span class="lab"><font color="Green" title="E81">A261</font></span></a>, <a class="txt" href="#E11:43_14_2"><span class="lab"><font color="Green" title="E93">A273</font></span></a>, <a class="txt" href="#E1:43_14_2_4_1"><span class="lab"><font color="Green" title="E95">A275</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_14_2_4_2"><span class="kw">suppose </span></a><a NAME="E1:43_14_2_4_2"/><span class="lab"><font color="Green" title="E95">A277</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">k</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="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/><div class="add"><a NAME="E2:43_14_2_4_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c7">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> <font color="Maroon" title="c5">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E47:43"><span class="lab"><font color="Green" title="E77">A257</font></span></a>, <a class="txt" href="#E7:43_14"><span class="lab"><font color="Green" title="E84">A264</font></span></a>, <a class="txt" href="#E4:43_14_2"><span class="lab"><font color="Green" title="E89">A269</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/><a NAME="E3:43_14_2_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E96">A278</font></span>: 
 ex <font color="Olive" title="b1">p2</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font> st <br/>( <font color="Olive" title="b1">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> &amp; <font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</a></span>;<br/><a NAME="E4:43_14_2_4_2"/>
<font color="Maroon" title="c10">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="E5:43_14_2_4_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A279</font></span>: 
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c10">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:43"><span class="lab"><font color="Green" title="E40">A7</font></span></a>, <a class="txt" href="#E3:43_14_2"><span class="lab"><font color="Green" title="E88">A268</font></span></a>, <a class="txt" href="#E11:43_14_2"><span class="lab"><font color="Green" title="E93">A273</font></span></a>, <a class="txt" href="#E1:43_14_2_4_2"><span class="lab"><font color="Green" title="E95">A277</font></span></a>, <a class="txt" href="#E3:43_14_2_4_2"><span class="lab"><font color="Green" title="E96">A278</font></span></a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_14_2_4_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_14_2_4_2_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="E1:43_14_2_4_2_1_1"/>
( <font color="Maroon" title="c10">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">j</font> or <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_14_2_4_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:43_14_2_4_2_1_1_1"/><span class="lab"><font color="Green" title="E98">A280</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">j</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="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/><div class="add"><a NAME="E2:43_14_2_4_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A281</font></span>: 
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_14"><span class="lab"><font color="Green" title="E79">A259</font></span></a>, <a class="txt" href="#E2:43_14"><span class="lab"><font color="Green" title="E80">A260</font></span></a>, <a class="txt" href="#E3:43_14"><span class="lab"><font color="Green" title="E81">A261</font></span></a>, <a class="txt" href="#E5:43_14_2_4_2"><span class="lab"><font color="Green" title="E97">A279</font></span></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">Th25</a></span>;<br/><a NAME="E3:43_14_2_4_2_1_1_1"/><span class="lab"><font color="Green" title="E100">A282</font></span>: 
<span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_14"><span class="lab"><font color="Green" title="E82">A262</font></span></a>, <a class="txt" href="#E2:43_14_2"><span class="lab"><font color="Green" title="E87">A267</font></span></a>, <a class="txt" href="#E3:43_14_2"><span class="lab"><font color="Green" title="E88">A268</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/><span class="lab"><font color="Green" title="E101">A283</font></span>: <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_14_2"><span class="lab"><font color="Green" title="E88">A268</font></span></a>, <a class="txt" href="#E7:43_14_2"><span class="lab"><font color="Green" title="E91">A271</font></span></a>, <a class="txt" href="#E12:43_14_2"><span class="lab"><font color="Green" title="E94">A274</font></span></a>, <a class="txt" href="#E1:43_14_2_4_2"><span class="lab"><font color="Green" title="E95">A277</font></span></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">Th25</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></span>
;<br/><span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:43_14"><span class="lab"><font color="Green" title="E83">A263</font></span></a>, <a class="txt" href="#E2:43_14_2"><span class="lab"><font color="Green" title="E87">A267</font></span></a>, <a class="txt" href="#E3:43_14_2"><span class="lab"><font color="Green" title="E88">A268</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_14_2"><span class="lab"><font color="Green" title="E87">A267</font></span></a>, <a class="txt" href="#E3:43_14_2"><span class="lab"><font color="Green" title="E88">A268</font></span></a>, <a class="txt" href="#E1:43_14_2_4_2_1_1_1"><span class="lab"><font color="Green" title="E98">A280</font></span></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">Th25</a></span>
;<br/><a NAME="E6:43_14_2_4_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_14"><span class="lab"><font color="Green" title="E82">A262</font></span></a>, <a class="txt" href="#E6:43_14"><span class="lab"><font color="Green" title="E83">A263</font></span></a>, <a class="txt" href="#E2:43_14_2"><span class="lab"><font color="Green" title="E87">A267</font></span></a>, <a class="txt" href="#E3:43_14_2"><span class="lab"><font color="Green" title="E88">A268</font></span></a>, <a class="txt" href="#E3:43_14_2_4_2_1_1_1"><span class="lab"><font color="Green" title="E100">A282</font></span></a>, <a class="txt" href="#E4:43_14_2_4_2_1_1_1"><span class="lab"><font color="Green" title="E101">A283</font></span></a>, <a class="ref" href="matrix14.html#T5" target="_self" title="MATRIX14:th.5">Th5</a></span>;<br/><span class="kw">hence </span><a NAME="E7:43_14_2_4_2_1_1_1"/>
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_14_2_4_2_1_1_1"><span class="lab"><font color="Green" title="E99">A281</font></span></a>, <a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</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="43_14_2_4_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:43_14_2_4_2_1_1_2"/><span class="lab"><font color="Green" title="E98">A284</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</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="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/><div class="add"><a NAME="E2:43_14_2_4_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A285</font></span>: 
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_14_2"><span class="lab"><font color="Green" title="E87">A267</font></span></a>, <a class="txt" href="#E3:43_14_2"><span class="lab"><font color="Green" title="E88">A268</font></span></a>, <a class="txt" href="#E5:43_14_2_4_2"><span class="lab"><font color="Green" title="E97">A279</font></span></a>, <a class="ref" href="matrix14.html#T24" target="_self" title="MATRIX14:th.24">Th24</a></span>;<br/><a NAME="E3:43_14_2_4_2_1_1_2"/><span class="lab"><font color="Green" title="E100">A286</font></span>: 
<span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_14"><span class="lab"><font color="Green" title="E82">A262</font></span></a>, <a class="txt" href="#E2:43_14_2"><span class="lab"><font color="Green" title="E87">A267</font></span></a>, <a class="txt" href="#E3:43_14_2"><span class="lab"><font color="Green" title="E88">A268</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/><span class="lab"><font color="Green" title="E101">A287</font></span>: <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_14_2"><span class="lab"><font color="Green" title="E88">A268</font></span></a>, <a class="txt" href="#E7:43_14_2"><span class="lab"><font color="Green" title="E91">A271</font></span></a>, <a class="txt" href="#E12:43_14_2"><span class="lab"><font color="Green" title="E94">A274</font></span></a>, <a class="txt" href="#E1:43_14_2_4_2"><span class="lab"><font color="Green" title="E95">A277</font></span></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">Th25</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></span>
;<br/><span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:43_14"><span class="lab"><font color="Green" title="E83">A263</font></span></a>, <a class="txt" href="#E2:43_14_2"><span class="lab"><font color="Green" title="E87">A267</font></span></a>, <a class="txt" href="#E3:43_14_2"><span class="lab"><font color="Green" title="E88">A268</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>
<br/>.= 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_14"><span class="lab"><font color="Green" title="E79">A259</font></span></a>, <a class="txt" href="#E2:43_14"><span class="lab"><font color="Green" title="E80">A260</font></span></a>, <a class="txt" href="#E3:43_14"><span class="lab"><font color="Green" title="E81">A261</font></span></a>, <a class="txt" href="#E1:43_14_2_4_2_1_1_2"><span class="lab"><font color="Green" title="E98">A284</font></span></a>, <a class="ref" href="matrix14.html#T24" target="_self" title="MATRIX14:th.24">Th24</a></span>
;<br/><a NAME="E6:43_14_2_4_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_14"><span class="lab"><font color="Green" title="E82">A262</font></span></a>, <a class="txt" href="#E6:43_14"><span class="lab"><font color="Green" title="E83">A263</font></span></a>, <a class="txt" href="#E2:43_14_2"><span class="lab"><font color="Green" title="E87">A267</font></span></a>, <a class="txt" href="#E3:43_14_2"><span class="lab"><font color="Green" title="E88">A268</font></span></a>, <a class="txt" href="#E3:43_14_2_4_2_1_1_2"><span class="lab"><font color="Green" title="E100">A286</font></span></a>, <a class="txt" href="#E4:43_14_2_4_2_1_1_2"><span class="lab"><font color="Green" title="E101">A287</font></span></a>, <a class="ref" href="matrix14.html#T5" target="_self" title="MATRIX14:th.5">Th5</a></span>;<br/><span class="kw">hence </span><a NAME="E7:43_14_2_4_2_1_1_2"/>
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_14_2_4_2_1_1_2"><span class="lab"><font color="Green" title="E99">A285</font></span></a>, <a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</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><span class="kw">hence </span><a NAME="E7:43_14_2_4_2"/>
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 ; <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 NAME="E9:43_14"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">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> <font color="Maroon" title="c4">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_14"><span class="lab"><font color="Green" title="E79">A259</font></span></a>, <a class="txt" href="#E2:43_14"><span class="lab"><font color="Green" title="E80">A260</font></span></a>, <a class="txt" href="#E3:43_14"><span class="lab"><font color="Green" title="E81">A261</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">p1</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font><span class="kw"> such that </span><br/><a NAME="E11:43_14"/><span class="lab"><font color="Green" title="E86">A288</font></span>: 
<font color="Maroon" title="c10">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
 <span class="kw">and </span><br/><a NAME="E12:43_14"/><span class="lab"><font color="Green" title="E87">A289</font></span>: 
<font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</a></span>;<br/>
<a NAME="E13:43_14"/>
<font color="Maroon" title="c10">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43"><span class="lab"><font color="Green" title="E37">A4</font></span></a>, <a class="txt" href="#E1:43_14"><span class="lab"><font color="Green" title="E79">A259</font></span></a>, <a class="txt" href="#E2:43_14"><span class="lab"><font color="Green" title="E80">A260</font></span></a>, <a class="txt" href="#E11:43_14"><span class="lab"><font color="Green" title="E86">A288</font></span></a></span>;<br/>
<a NAME="E14:43_14"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A290</font></span>: 
<font color="Maroon" title="c10">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_14"><span class="lab"><font color="Green" title="E79">A259</font></span></a>, <a class="txt" href="#E2:43_14"><span class="lab"><font color="Green" title="E80">A260</font></span></a>, <a class="txt" href="#E3:43_14"><span class="lab"><font color="Green" title="E81">A261</font></span></a>, <a class="ref" href="matrix14.html#T24" target="_self" title="MATRIX14:th.24">Th24</a></span>;<br/>
<span class="lab"><font color="Green" title="E89">A291</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c7">j</font></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">Q</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></span>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></span>
;<br/>
<a NAME="E16:43_14"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p4">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_14"><span class="lab"><font color="Green" title="E82">A262</font></span></a>, <a class="txt" href="#E6:43_14"><span class="lab"><font color="Green" title="E83">A263</font></span></a>, <a class="ref" href="matrix14.html#T2" target="_self" title="MATRIX14:th.2">Th2</a></span>;<br/>
<a NAME="E17:43_14"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A292</font></span>: 
 <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:43_14"><span class="lab"><font color="Green" title="E89">A291</font></span></a>, <a class="txt" href="#E8:43_14"><span class="lab"><font color="Green" title="E85">A265</font></span></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></span>;<br/>
<a NAME="E18:43_14"/><span class="lab"><font color="Green" title="E91">A293</font></span>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></span>;<br/>
<a NAME="E19:43_14"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A294</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E93">A295</font></span>: <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:43"><span class="lab"><font color="Green" title="E68">A219</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>
<br/>.= 
<font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:43"><span class="lab"><font color="Green" title="E75">A226</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>
;<br/>
<span class="lab"><font color="Green" title="E94">A296</font></span>: <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">j</font> = 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_14"><span class="lab"><font color="Green" title="E79">A259</font></span></a>, <a class="txt" href="#E2:43_14"><span class="lab"><font color="Green" title="E80">A260</font></span></a>, <a class="txt" href="#E3:43_14"><span class="lab"><font color="Green" title="E81">A261</font></span></a>, <a class="txt" href="#E19:43_14"><span class="lab"><font color="Green" title="E92">A294</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>
<br/>.= 
<font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:43_14"><span class="lab"><font color="Green" title="E84">A264</font></span></a>, <a class="txt" href="#E18:43_14"><span class="lab"><font color="Green" title="E91">A293</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>
;<br/>
<a NAME="E22:43_14"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">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="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_14"><span class="lab"><font color="Green" title="E79">A259</font></span></a>, <a class="txt" href="#E2:43_14"><span class="lab"><font color="Green" title="E80">A260</font></span></a>, <a class="txt" href="#E3:43_14"><span class="lab"><font color="Green" title="E81">A261</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E95">A297</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> = 
<span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:43"><span class="lab"><font color="Green" title="E64">A202</font></span></a>, <a class="txt" href="#E18:43_14"><span class="lab"><font color="Green" title="E91">A293</font></span></a>, <a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a></span>
<br/>.= 
<span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p4">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_14"><span class="lab"><font color="Green" title="E82">A262</font></span></a>, <a class="txt" href="#E6:43_14"><span class="lab"><font color="Green" title="E83">A263</font></span></a>, <a class="txt" href="#E17:43_14"><span class="lab"><font color="Green" title="E90">A292</font></span></a>, <a class="txt" href="#E19:43_14"><span class="lab"><font color="Green" title="E92">A294</font></span></a>, <a class="ref" href="matrix14.html#T11" target="_self" title="MATRIX14:th.11">Th11</a></span>
<br/>.= 
<span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix14.html#T6" target="_self" title="MATRIX14:th.6">Th6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:43"><span class="lab"><font color="Green" title="E65">A203</font></span></a>, <a class="txt" href="#E19:43_14"><span class="lab"><font color="Green" title="E92">A294</font></span></a>, <a class="ref" href="matrix14.html#T10" target="_self" title="MATRIX14:th.10">Th10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,1</span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:43"><span class="lab"><font color="Green" title="E68">A219</font></span></a>, <a class="txt" href="#E19:43_14"><span class="lab"><font color="Green" title="E92">A294</font></span></a>, <a class="txt" href="#E20:43_14"><span class="lab"><font color="Green" title="E93">A295</font></span></a>, <a class="ref" href="matrix14.html#T35" target="_self" title="MATRIX14:th.35">Th35</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,1</span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T41" title="VECTSP_1:th.41">VECTSP_1:41</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_14"><span class="lab"><font color="Green" title="E79">A259</font></span></a>, <a class="txt" href="#E2:43_14"><span class="lab"><font color="Green" title="E80">A260</font></span></a>, <a class="txt" href="#E3:43_14"><span class="lab"><font color="Green" title="E81">A261</font></span></a>, <a class="txt" href="#E19:43_14"><span class="lab"><font color="Green" title="E92">A294</font></span></a>, <a class="txt" href="#E21:43_14"><span class="lab"><font color="Green" title="E94">A296</font></span></a>, <a class="ref" href="matrix14.html#T35" target="_self" title="MATRIX14:th.35">Th35</a></span>
;<br/>
<a NAME="E24:43_14"/><span class="lab"><font color="Green" title="E96">A298</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_14"><span class="lab"><font color="Green" title="E79">A259</font></span></a>, <a class="txt" href="#E2:43_14"><span class="lab"><font color="Green" title="E80">A260</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E25:43_14"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">i</font>,1</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> <font color="Maroon" title="c4">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:43"><span class="lab"><font color="Green" title="E41">A8</font></span></a>, <a class="txt" href="#E1:43_14"><span class="lab"><font color="Green" title="E79">A259</font></span></a>, <a class="txt" href="#E2:43_14"><span class="lab"><font color="Green" title="E80">A260</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">p2</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font><span class="kw"> such that </span><br/><a NAME="E27:43_14"/><span class="lab"><font color="Green" title="E97">A299</font></span>: 
<font color="Maroon" title="c11">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
 <span class="kw">and </span><br/><a NAME="E28:43_14"/><span class="lab"><font color="Green" title="E98">A300</font></span>: 
<font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</a></span>;<br/>
<a NAME="E29:43_14"/>
<font color="Maroon" title="c11">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43"><span class="lab"><font color="Green" title="E37">A4</font></span></a>, <a class="txt" href="#E1:43_14"><span class="lab"><font color="Green" title="E79">A259</font></span></a>, <a class="txt" href="#E2:43_14"><span class="lab"><font color="Green" title="E80">A260</font></span></a>, <a class="txt" href="#E27:43_14"><span class="lab"><font color="Green" title="E97">A299</font></span></a></span>;<br/>
<span class="kw">hence </span><span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> = 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_14"><span class="lab"><font color="Green" title="E79">A259</font></span></a>, <a class="txt" href="#E24:43_14"><span class="lab"><font color="Green" title="E96">A298</font></span></a>, <a class="txt" href="#E23:43_14"><span class="lab"><font color="Green" title="E95">A297</font></span></a>, <a class="txt" href="#E28:43_14"><span class="lab"><font color="Green" title="E98">A300</font></span></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">Th25</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:43_14"><span class="lab"><font color="Green" title="E87">A289</font></span></a>, <a class="txt" href="#E14:43_14"><span class="lab"><font color="Green" title="E88">A290</font></span></a>, <a class="ref" href="rlvect_1.html#T25" title="RLVECT_1:th.25">RLVECT_1:25</a></span>
<br/>.= 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></span>
;<br/> <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>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">Q</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></span>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></span>
;<br/>
<a NAME="E50:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A301</font></span>: 
 <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">Q</font>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:43"><span class="lab"><font color="Green" title="E66">A204</font></span></a>, <a class="txt" href="#E37:43"><span class="lab"><font color="Green" title="E67">A205</font></span></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></span>;<br/>
<a NAME="E51:43"/><span class="lab"><font color="Green" title="E80">A302</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="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<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>,<span class="p2">(<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></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></span>;<br/>
<a NAME="E52:43"/><span class="lab"><font color="Green" title="E81">A303</font></span>: 
for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j</font> &amp; <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Olive" title="b1">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">j</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_16"><span class="kw">proof </span></a><div class="add">

<span class="lab"><font color="Green" title="E82">A304</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixr1.html#T16" title="MATRIXR1:th.16">MATRIXR1:16</a></span>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">Th23</a></span>
;<br/>
<span class="kw">let </span><font color="Maroon" title="c6">i</font>, <font color="Maroon" title="c7">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">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#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">i</font> &amp; <font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">j</font> &amp; <font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c6">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">j</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:43_16"/><span class="lab"><font color="Green" title="E83">A305</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">i</font>
 <span class="kw">and </span><br/><a NAME="E3:43_16"/><span class="lab"><font color="Green" title="E84">A306</font></span>: 
<font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">and </span><br/><a NAME="E4:43_16"/><span class="lab"><font color="Green" title="E85">A307</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">j</font>
 <span class="kw">and </span><br/><a NAME="E5:43_16"/><span class="lab"><font color="Green" title="E86">A308</font></span>: 
<font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">and </span><br/><a NAME="E6:43_16"/><span class="lab"><font color="Green" title="E87">A309</font></span>: 
<font color="Maroon" title="c6">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">j</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="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span><br/>

<a NAME="E7:43_16"/><span class="lab"><font color="Green" title="E88">A310</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">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="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_16"><span class="lab"><font color="Green" title="E83">A305</font></span></a>, <a class="txt" href="#E3:43_16"><span class="lab"><font color="Green" title="E84">A306</font></span></a>, <a class="txt" href="#E4:43_16"><span class="lab"><font color="Green" title="E85">A307</font></span></a>, <a class="txt" href="#E5:43_16"><span class="lab"><font color="Green" title="E86">A308</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E8:43_16"/><span class="lab"><font color="Green" title="E89">A311</font></span>: 
<font color="Maroon" title="c7">j</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> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_16"><span class="lab"><font color="Green" title="E85">A307</font></span></a>, <a class="txt" href="#E5:43_16"><span class="lab"><font color="Green" title="E86">A308</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<span class="lab"><font color="Green" title="E90">A312</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c7">j</font></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">Q</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></span>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></span>
;<br/>
<a NAME="E10:43_16"/><span class="lab"><font color="Green" title="E91">A313</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">i</font>,1</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> <font color="Maroon" title="c4">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:43"><span class="lab"><font color="Green" title="E41">A8</font></span></a>, <a class="txt" href="#E2:43_16"><span class="lab"><font color="Green" title="E83">A305</font></span></a>, <a class="txt" href="#E3:43_16"><span class="lab"><font color="Green" title="E84">A306</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E11:43_16"/><span class="lab"><font color="Green" title="E92">A314</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">Th23</a></span>;<br/>
<span class="lab"><font color="Green" title="E93">A315</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix14.html#T6" target="_self" title="MATRIX14:th.6">Th6</a></span>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixr1.html#T16" title="MATRIXR1:th.16">MATRIXR1:16</a></span>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">Th23</a></span>
;<br/>
<a NAME="E13:43_16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E94">A316</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p4">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:43_16"><span class="lab"><font color="Green" title="E92">A314</font></span></a>, <a class="ref" href="matrix14.html#T2" target="_self" title="MATRIX14:th.2">Th2</a></span>;<br/>
<a NAME="E14:43_16"/><span class="lab"><font color="Green" title="E95">A317</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">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> <font color="Maroon" title="c4">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_16"><span class="lab"><font color="Green" title="E83">A305</font></span></a>, <a class="txt" href="#E3:43_16"><span class="lab"><font color="Green" title="E84">A306</font></span></a>, <a class="txt" href="#E4:43_16"><span class="lab"><font color="Green" title="E85">A307</font></span></a>, <a class="txt" href="#E5:43_16"><span class="lab"><font color="Green" title="E86">A308</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_16_4"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_16_4_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="E1:43_16_4_1"/>
( <font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 or <font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_16_4_1_1"><span class="kw">suppose </span></a><a NAME="E1:43_16_4_1_1"/><span class="lab"><font color="Green" title="E96">A318</font></span>: 
<font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ; <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="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c8">p</font> =  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c7">j</font>, <font color="Maroon" title="c9">q</font> = <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font> ;<br/><a NAME="E3:43_16_4_1_1"/>
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> <font color="Maroon" title="c1">n</font> holds <br/><font color="Maroon" title="c8">p</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="c9">q</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="43_16_4_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">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="c10">k</font> &amp; <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> implies <font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:43_16_4_1_1_1"/><span class="lab"><font color="Green" title="E97">A319</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font>
 <span class="kw">and </span><br/><a NAME="E2:43_16_4_1_1_1"/><span class="lab"><font color="Green" title="E98">A320</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</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="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/>

<a NAME="E3:43_16_4_1_1_1"/><span class="lab"><font color="Green" title="E99">A321</font></span>: 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:43"><span class="lab"><font color="Green" title="E65">A203</font></span></a>, <a class="txt" href="#E1:43_16_4_1_1_1"><span class="lab"><font color="Green" title="E97">A319</font></span></a>, <a class="txt" href="#E2:43_16_4_1_1_1"><span class="lab"><font color="Green" title="E98">A320</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/>
<span class="lab"><font color="Green" title="E100">A322</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixr1.html#T16" title="MATRIXR1:th.16">MATRIXR1:16</a></span>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">Th23</a></span>
;<br/>
<a NAME="E5:43_16_4_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A323</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_16_4_1_1_1"><span class="lab"><font color="Green" title="E97">A319</font></span></a>, <a class="txt" href="#E2:43_16_4_1_1_1"><span class="lab"><font color="Green" title="E98">A320</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<span class="lab"><font color="Green" title="E102">A324</font></span>: <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix14.html#T6" target="_self" title="MATRIX14:th.6">Th6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43_16_4_1_1_1"><span class="lab"><font color="Green" title="E101">A323</font></span></a>, <a class="txt" href="#E3:43_16_4_1_1_1"><span class="lab"><font color="Green" title="E99">A321</font></span></a>, <a class="ref" href="fvsum_1.html#T62" title="FVSUM_1:th.62">FVSUM_1:62</a></span>
;<br/>
<a NAME="E7:43_16_4_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">Th23</a></span>;<br/>
<a NAME="E8:43_16_4_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A325</font></span>: 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_16_4_1_1_1"><span class="lab"><font color="Green" title="E97">A319</font></span></a>, <a class="txt" href="#E2:43_16_4_1_1_1"><span class="lab"><font color="Green" title="E98">A320</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/>
<a NAME="E9:43_16_4_1_1_1"/>
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">Q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:43"><span class="lab"><font color="Green" title="E64">A202</font></span></a>, <a class="txt" href="#E1:43_16_4_1_1_1"><span class="lab"><font color="Green" title="E97">A319</font></span></a>, <a class="txt" href="#E2:43_16_4_1_1_1"><span class="lab"><font color="Green" title="E98">A320</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E10:43_16_4_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A326</font></span>: 
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c7">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_16_4_1_1_1_3"><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="E1:43_16_4_1_1_1_3"/>
( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</font> or 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">k</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_16_4_1_1_1"><span class="lab"><font color="Green" title="E97">A319</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_16_4_1_1_1_3_1"><span class="kw">suppose </span></a><a NAME="E1:43_16_4_1_1_1_3_1"/><span class="lab"><font color="Green" title="E105">A327</font></span>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</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="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/><div class="add"><a NAME="E2:43_16_4_1_1_1_3_1"/>
<font color="Maroon" title="c10">k</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> <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_16_4_1_1_1"><span class="lab"><font color="Green" title="E98">A320</font></span></a>, <a class="txt" href="#E4:43_16_4_1_1_1"><span class="lab"><font color="Green" title="E100">A322</font></span></a>, <a class="ref" href="matrix14.html#T6" target="_self" title="MATRIX14:th.6">Th6</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E106">A328</font></span>: <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_16_4_1_1_1"><span class="lab"><font color="Green" title="E97">A319</font></span></a>, <a class="txt" href="#E6:43_16_4_1_1_1"><span class="lab"><font color="Green" title="E102">A324</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:43"><span class="lab"><font color="Green" title="E68">A219</font></span></a>, <a class="txt" href="#E8:43_16_4_1_1_1"><span class="lab"><font color="Green" title="E103">A325</font></span></a>, <a class="txt" href="#E1:43_16_4_1_1_1_3_1"><span class="lab"><font color="Green" title="E105">A327</font></span></a>, <a class="ref" href="matrix14.html#T24" target="_self" title="MATRIX14:th.24">Th24</a></span>
;<br/><span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:43_16"><span class="lab"><font color="Green" title="E92">A314</font></span></a>, <a class="txt" href="#E1:43_16_4_1_1_1"><span class="lab"><font color="Green" title="E97">A319</font></span></a>, <a class="txt" href="#E2:43_16_4_1_1_1"><span class="lab"><font color="Green" title="E98">A320</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_16_4_1_1"><span class="lab"><font color="Green" title="E96">A318</font></span></a>, <a class="txt" href="#E2:43_16_4_1_1_1"><span class="lab"><font color="Green" title="E98">A320</font></span></a>, <a class="txt" href="#E1:43_16_4_1_1_1_3_1"><span class="lab"><font color="Green" title="E105">A327</font></span></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">Th25</a></span>
;<br/><span class="kw">then </span><font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:43_16"><span class="lab"><font color="Green" title="E93">A315</font></span></a>, <a class="txt" href="#E11:43_16"><span class="lab"><font color="Green" title="E92">A314</font></span></a>, <a class="txt" href="#E1:43_16_4_1_1_1"><span class="lab"><font color="Green" title="E97">A319</font></span></a>, <a class="txt" href="#E2:43_16_4_1_1_1"><span class="lab"><font color="Green" title="E98">A320</font></span></a>, <a class="txt" href="#E3:43_16_4_1_1_1_3_1"><span class="lab"><font color="Green" title="E106">A328</font></span></a>, <a class="ref" href="matrix14.html#T5" target="_self" title="MATRIX14:th.5">Th5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c2">K</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></span>
<br/>.= 
 <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D5" title="GROUP_1:def.5">GROUP_1:def 5</a></span>
;<br/><span class="kw">hence </span><a NAME="E6:43_16_4_1_1_1_3_1"/>
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:43"><span class="lab"><font color="Green" title="E39">A6</font></span></a>, <a class="txt" href="#E5:43_16"><span class="lab"><font color="Green" title="E86">A308</font></span></a>, <a class="txt" href="#E1:43_16_4_1_1"><span class="lab"><font color="Green" title="E96">A318</font></span></a>, <a class="txt" href="#E10:43_16_4_1_1_1"><span class="lab"><font color="Green" title="E104">A326</font></span></a>, <a class="txt" href="#E1:43_16_4_1_1_1_3_1"><span class="lab"><font color="Green" title="E105">A327</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_16_4_1_1_1_3_2"><span class="kw">suppose </span></a><a NAME="E1:43_16_4_1_1_1_3_2"/><span class="lab"><font color="Green" title="E105">A329</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">k</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="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/><div class="add"><a NAME="E2:43_16_4_1_1_1_3_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c7">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> <font color="Maroon" title="c5">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_16"><span class="lab"><font color="Green" title="E85">A307</font></span></a>, <a class="txt" href="#E5:43_16"><span class="lab"><font color="Green" title="E86">A308</font></span></a>, <a class="txt" href="#E1:43_16_4_1_1_1"><span class="lab"><font color="Green" title="E97">A319</font></span></a>, <a class="txt" href="#E2:43_16_4_1_1_1"><span class="lab"><font color="Green" title="E98">A320</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E3:43_16_4_1_1_1_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E106">A330</font></span>: 
 ex <font color="Olive" title="b1">p2</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font> st <br/>( <font color="Olive" title="b1">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> &amp; <font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</a></span>;<br/><a NAME="E4:43_16_4_1_1_1_3_2"/>
<font color="Maroon" title="c10">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="E5:43_16_4_1_1_1_3_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E107">A331</font></span>: 
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c10">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:43"><span class="lab"><font color="Green" title="E40">A7</font></span></a>, <a class="txt" href="#E2:43_16_4_1_1_1"><span class="lab"><font color="Green" title="E98">A320</font></span></a>, <a class="txt" href="#E10:43_16_4_1_1_1"><span class="lab"><font color="Green" title="E104">A326</font></span></a>, <a class="txt" href="#E1:43_16_4_1_1_1_3_2"><span class="lab"><font color="Green" title="E105">A329</font></span></a>, <a class="txt" href="#E3:43_16_4_1_1_1_3_2"><span class="lab"><font color="Green" title="E106">A330</font></span></a></span>;<br/><a NAME="E6:43_16_4_1_1_1_3_2"/><span class="lab"><font color="Green" title="E108">A332</font></span>: 
<font color="Maroon" title="c10">k</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> <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_16_4_1_1_1"><span class="lab"><font color="Green" title="E98">A320</font></span></a>, <a class="txt" href="#E4:43_16_4_1_1_1"><span class="lab"><font color="Green" title="E100">A322</font></span></a>, <a class="ref" href="matrix14.html#T6" target="_self" title="MATRIX14:th.6">Th6</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_16_4_1_1_1_3_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_16_4_1_1_1_3_2_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="E1:43_16_4_1_1_1_3_2_1_1"/>
( <font color="Maroon" title="c10">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">j</font> or <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_16_4_1_1_1_3_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:43_16_4_1_1_1_3_2_1_1_1"/><span class="lab"><font color="Green" title="E109">A333</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">j</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="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/><div class="add"><span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:43_16"><span class="lab"><font color="Green" title="E92">A314</font></span></a>, <a class="txt" href="#E1:43_16_4_1_1_1"><span class="lab"><font color="Green" title="E97">A319</font></span></a>, <a class="txt" href="#E2:43_16_4_1_1_1"><span class="lab"><font color="Green" title="E98">A320</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_16_4_1_1_1"><span class="lab"><font color="Green" title="E97">A319</font></span></a>, <a class="txt" href="#E2:43_16_4_1_1_1"><span class="lab"><font color="Green" title="E98">A320</font></span></a>, <a class="txt" href="#E1:43_16_4_1_1_1_3_2_1_1_1"><span class="lab"><font color="Green" title="E109">A333</font></span></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">Th25</a></span>
;<br/><a NAME="E3:43_16_4_1_1_1_3_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p4">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:43_16"><span class="lab"><font color="Green" title="E93">A315</font></span></a>, <a class="txt" href="#E11:43_16"><span class="lab"><font color="Green" title="E92">A314</font></span></a>, <a class="txt" href="#E1:43_16_4_1_1_1"><span class="lab"><font color="Green" title="E97">A319</font></span></a>, <a class="txt" href="#E2:43_16_4_1_1_1"><span class="lab"><font color="Green" title="E98">A320</font></span></a>, <a class="ref" href="matrix14.html#T5" target="_self" title="MATRIX14:th.5">Th5</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E110">A334</font></span>: <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_16_4_1_1_1"><span class="lab"><font color="Green" title="E97">A319</font></span></a>, <a class="txt" href="#E6:43_16_4_1_1_1_3_2"><span class="lab"><font color="Green" title="E108">A332</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>
;<br/><span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_16_4_1_1_1"><span class="lab"><font color="Green" title="E98">A320</font></span></a>, <a class="txt" href="#E6:43_16_4_1_1_1"><span class="lab"><font color="Green" title="E102">A324</font></span></a>, <a class="txt" href="#E8:43_16_4_1_1_1"><span class="lab"><font color="Green" title="E103">A325</font></span></a>, <a class="txt" href="#E1:43_16_4_1_1_1_3_2"><span class="lab"><font color="Green" title="E105">A329</font></span></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">Th25</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></span>
;<br/><span class="kw">hence </span><a NAME="E6:43_16_4_1_1_1_3_2_1_1_1"/>
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_16"><span class="lab"><font color="Green" title="E85">A307</font></span></a>, <a class="txt" href="#E5:43_16"><span class="lab"><font color="Green" title="E86">A308</font></span></a>, <a class="txt" href="#E5:43_16_4_1_1_1_3_2"><span class="lab"><font color="Green" title="E107">A331</font></span></a>, <a class="txt" href="#E1:43_16_4_1_1_1_3_2_1_1_1"><span class="lab"><font color="Green" title="E109">A333</font></span></a>, <a class="txt" href="#E4:43_16_4_1_1_1_3_2_1_1_1"><span class="lab"><font color="Green" title="E110">A334</font></span></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">Th25</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="43_16_4_1_1_1_3_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:43_16_4_1_1_1_3_2_1_1_2"/><span class="lab"><font color="Green" title="E109">A335</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</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="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/><div class="add"><a NAME="E2:43_16_4_1_1_1_3_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E110">A336</font></span>: 
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_16_4_1_1_1"><span class="lab"><font color="Green" title="E97">A319</font></span></a>, <a class="txt" href="#E2:43_16_4_1_1_1"><span class="lab"><font color="Green" title="E98">A320</font></span></a>, <a class="txt" href="#E5:43_16_4_1_1_1_3_2"><span class="lab"><font color="Green" title="E107">A331</font></span></a>, <a class="ref" href="matrix14.html#T24" target="_self" title="MATRIX14:th.24">Th24</a></span>;<br/><a NAME="E3:43_16_4_1_1_1_3_2_1_1_2"/><span class="lab"><font color="Green" title="E111">A337</font></span>: 
<span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:43_16"><span class="lab"><font color="Green" title="E93">A315</font></span></a>, <a class="txt" href="#E1:43_16_4_1_1_1"><span class="lab"><font color="Green" title="E97">A319</font></span></a>, <a class="txt" href="#E2:43_16_4_1_1_1"><span class="lab"><font color="Green" title="E98">A320</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/><span class="lab"><font color="Green" title="E112">A338</font></span>: <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_16_4_1_1_1"><span class="lab"><font color="Green" title="E98">A320</font></span></a>, <a class="txt" href="#E3:43_16_4_1_1_1"><span class="lab"><font color="Green" title="E99">A321</font></span></a>, <a class="txt" href="#E6:43_16_4_1_1_1"><span class="lab"><font color="Green" title="E102">A324</font></span></a>, <a class="txt" href="#E1:43_16_4_1_1_1_3_2"><span class="lab"><font color="Green" title="E105">A329</font></span></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">Th25</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></span>
;<br/><span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> = 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:43_16"><span class="lab"><font color="Green" title="E92">A314</font></span></a>, <a class="txt" href="#E1:43_16_4_1_1_1"><span class="lab"><font color="Green" title="E97">A319</font></span></a>, <a class="txt" href="#E2:43_16_4_1_1_1"><span class="lab"><font color="Green" title="E98">A320</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>
<br/>.= 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_16_4_1_1_1"><span class="lab"><font color="Green" title="E97">A319</font></span></a>, <a class="txt" href="#E2:43_16_4_1_1_1"><span class="lab"><font color="Green" title="E98">A320</font></span></a>, <a class="txt" href="#E1:43_16_4_1_1_1_3_2_1_1_2"><span class="lab"><font color="Green" title="E109">A335</font></span></a>, <a class="ref" href="matrix14.html#T24" target="_self" title="MATRIX14:th.24">Th24</a></span>
;<br/><a NAME="E6:43_16_4_1_1_1_3_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:43_16"><span class="lab"><font color="Green" title="E93">A315</font></span></a>, <a class="txt" href="#E11:43_16"><span class="lab"><font color="Green" title="E92">A314</font></span></a>, <a class="txt" href="#E1:43_16_4_1_1_1"><span class="lab"><font color="Green" title="E97">A319</font></span></a>, <a class="txt" href="#E2:43_16_4_1_1_1"><span class="lab"><font color="Green" title="E98">A320</font></span></a>, <a class="txt" href="#E3:43_16_4_1_1_1_3_2_1_1_2"><span class="lab"><font color="Green" title="E111">A337</font></span></a>, <a class="txt" href="#E4:43_16_4_1_1_1_3_2_1_1_2"><span class="lab"><font color="Green" title="E112">A338</font></span></a>, <a class="ref" href="matrix14.html#T5" target="_self" title="MATRIX14:th.5">Th5</a></span>;<br/><span class="kw">hence </span><a NAME="E7:43_16_4_1_1_1_3_2_1_1_2"/>
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_16_4_1_1_1_3_2_1_1_2"><span class="lab"><font color="Green" title="E110">A336</font></span></a>, <a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</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><span class="kw">hence </span><a NAME="E8:43_16_4_1_1_1_3_2"/>
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 ; <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 NAME="E4:43_16_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A339</font></span>: 
 <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K3" title="FVSUM_1:func.3">+</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:43_16"><span class="lab"><font color="Green" title="E90">A312</font></span></a>, <a class="txt" href="#E13:43_16"><span class="lab"><font color="Green" title="E94">A316</font></span></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></span>;<br/><a NAME="E5:43_16_4_1_1"/><span class="lab"><font color="Green" title="E98">A340</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_16"><span class="lab"><font color="Green" title="E83">A305</font></span></a>, <a class="txt" href="#E3:43_16"><span class="lab"><font color="Green" title="E84">A306</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E6:43_16_4_1_1"/><span class="lab"><font color="Green" title="E99">A341</font></span>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></span>;<br/><a NAME="E7:43_16_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A342</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E101">A343</font></span>: <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">j</font> = 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_16"><span class="lab"><font color="Green" title="E85">A307</font></span></a>, <a class="txt" href="#E5:43_16"><span class="lab"><font color="Green" title="E86">A308</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>
<br/>.= 
<font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:43_16"><span class="lab"><font color="Green" title="E89">A311</font></span></a>, <a class="txt" href="#E6:43_16_4_1_1"><span class="lab"><font color="Green" title="E99">A341</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>
;<br/><span class="lab"><font color="Green" title="E102">A344</font></span>: <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:43"><span class="lab"><font color="Green" title="E68">A219</font></span></a>, <a class="txt" href="#E7:43_16_4_1_1"><span class="lab"><font color="Green" title="E100">A342</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>
<br/>.= 
<font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:43"><span class="lab"><font color="Green" title="E75">A226</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>
;<br/><span class="lab"><font color="Green" title="E103">A345</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> = 
<span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:43"><span class="lab"><font color="Green" title="E64">A202</font></span></a>, <a class="txt" href="#E7:43_16"><span class="lab"><font color="Green" title="E88">A310</font></span></a>, <a class="txt" href="#E6:43_16_4_1_1"><span class="lab"><font color="Green" title="E99">A341</font></span></a>, <a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a></span>
<br/>.= 
<span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="fvsum_1.html#K5" title="FVSUM_1:func.5">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p4">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:43_16"><span class="lab"><font color="Green" title="E93">A315</font></span></a>, <a class="txt" href="#E11:43_16"><span class="lab"><font color="Green" title="E92">A314</font></span></a>, <a class="txt" href="#E4:43_16_4_1_1"><span class="lab"><font color="Green" title="E97">A339</font></span></a>, <a class="txt" href="#E7:43_16_4_1_1"><span class="lab"><font color="Green" title="E100">A342</font></span></a>, <a class="ref" href="matrix14.html#T11" target="_self" title="MATRIX14:th.11">Th11</a></span>
<br/>.= 
<span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p5">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix14.html#T6" target="_self" title="MATRIX14:th.6">Th6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p3">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:43"><span class="lab"><font color="Green" title="E65">A203</font></span></a>, <a class="txt" href="#E7:43_16_4_1_1"><span class="lab"><font color="Green" title="E100">A342</font></span></a>, <a class="ref" href="matrix14.html#T10" target="_self" title="MATRIX14:th.10">Th10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,1</span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:43"><span class="lab"><font color="Green" title="E68">A219</font></span></a>, <a class="txt" href="#E7:43_16_4_1_1"><span class="lab"><font color="Green" title="E100">A342</font></span></a>, <a class="txt" href="#E9:43_16_4_1_1"><span class="lab"><font color="Green" title="E102">A344</font></span></a>, <a class="ref" href="matrix14.html#T35" target="_self" title="MATRIX14:th.35">Th35</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,1</span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T41" title="VECTSP_1:th.41">VECTSP_1:41</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_16"><span class="lab"><font color="Green" title="E85">A307</font></span></a>, <a class="txt" href="#E5:43_16"><span class="lab"><font color="Green" title="E86">A308</font></span></a>, <a class="txt" href="#E7:43_16_4_1_1"><span class="lab"><font color="Green" title="E100">A342</font></span></a>, <a class="txt" href="#E8:43_16_4_1_1"><span class="lab"><font color="Green" title="E101">A343</font></span></a>, <a class="ref" href="matrix14.html#T35" target="_self" title="MATRIX14:th.35">Th35</a></span>
;<br/><span class="kw">consider </span><font color="Maroon" title="c10">p2</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font><span class="kw"> such that </span><br/><a NAME="E12:43_16_4_1_1"/><span class="lab"><font color="Green" title="E104">A346</font></span>: 
<font color="Maroon" title="c10">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
 <span class="kw">and </span><br/><a NAME="E13:43_16_4_1_1"/><span class="lab"><font color="Green" title="E105">A347</font></span>: 
<font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:43_16"><span class="lab"><font color="Green" title="E91">A313</font></span></a>, <a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c11">p1</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font><span class="kw"> such that </span><br/><a NAME="E15:43_16_4_1_1"/><span class="lab"><font color="Green" title="E106">A348</font></span>: 
<font color="Maroon" title="c11">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
 <span class="kw">and </span><br/><a NAME="E16:43_16_4_1_1"/><span class="lab"><font color="Green" title="E107">A349</font></span>: 
<font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:43_16"><span class="lab"><font color="Green" title="E95">A317</font></span></a>, <a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</a></span>;<br/><a NAME="E17:43_16_4_1_1"/>
<font color="Maroon" title="c11">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43"><span class="lab"><font color="Green" title="E37">A4</font></span></a>, <a class="txt" href="#E2:43_16"><span class="lab"><font color="Green" title="E83">A305</font></span></a>, <a class="txt" href="#E3:43_16"><span class="lab"><font color="Green" title="E84">A306</font></span></a>, <a class="txt" href="#E15:43_16_4_1_1"><span class="lab"><font color="Green" title="E106">A348</font></span></a></span>;<br/><a NAME="E18:43_16_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E108">A350</font></span>: 
<font color="Maroon" title="c11">p1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_16"><span class="lab"><font color="Green" title="E85">A307</font></span></a>, <a class="txt" href="#E5:43_16"><span class="lab"><font color="Green" title="E86">A308</font></span></a>, <a class="txt" href="#E6:43_16"><span class="lab"><font color="Green" title="E87">A309</font></span></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">Th25</a></span>;<br/><a NAME="E19:43_16_4_1_1"/>
<font color="Maroon" title="c10">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43"><span class="lab"><font color="Green" title="E37">A4</font></span></a>, <a class="txt" href="#E2:43_16"><span class="lab"><font color="Green" title="E83">A305</font></span></a>, <a class="txt" href="#E3:43_16"><span class="lab"><font color="Green" title="E84">A306</font></span></a>, <a class="txt" href="#E12:43_16_4_1_1"><span class="lab"><font color="Green" title="E104">A346</font></span></a></span>;<br/><span class="kw">then </span><span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> = 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p4">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_16"><span class="lab"><font color="Green" title="E83">A305</font></span></a>, <a class="txt" href="#E5:43_16_4_1_1"><span class="lab"><font color="Green" title="E98">A340</font></span></a>, <a class="txt" href="#E10:43_16_4_1_1"><span class="lab"><font color="Green" title="E103">A345</font></span></a>, <a class="txt" href="#E13:43_16_4_1_1"><span class="lab"><font color="Green" title="E105">A347</font></span></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">Th25</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p3">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span> <a href="algstr_0.html#K1" title="ALGSTR_0:func.1">+</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:43_16_4_1_1"><span class="lab"><font color="Green" title="E107">A349</font></span></a>, <a class="txt" href="#E18:43_16_4_1_1"><span class="lab"><font color="Green" title="E108">A350</font></span></a>, <a class="ref" href="rlvect_1.html#T25" title="RLVECT_1:th.25">RLVECT_1:25</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#T10" title="RLVECT_1:th.10">RLVECT_1:10</a></span>
;<br/><span class="kw">hence </span><a NAME="E21:43_16_4_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <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"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_16_4_1_2"><span class="kw">suppose </span></a><a NAME="E1:43_16_4_1_2"/><span class="lab"><font color="Green" title="E96">A351</font></span>: 
<font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 ; <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="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c8">p</font> =  <a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c7">j</font>, <font color="Maroon" title="c9">q</font> = <font color="Maroon" title="c3">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font> ;<br/><a NAME="E3:43_16_4_1_2"/><span class="lab"><font color="Green" title="E97">A352</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> <font color="Maroon" title="c1">n</font> holds <br/><font color="Maroon" title="c8">p</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="c9">q</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="43_16_4_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">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="c10">k</font> &amp; <font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> implies <font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:43_16_4_1_2_1"/><span class="lab"><font color="Green" title="E98">A353</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">k</font>
 <span class="kw">and </span><br/><a NAME="E2:43_16_4_1_2_1"/><span class="lab"><font color="Green" title="E99">A354</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</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="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/>

<a NAME="E3:43_16_4_1_2_1"/><span class="lab"><font color="Green" title="E100">A355</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">Q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:43"><span class="lab"><font color="Green" title="E64">A202</font></span></a>, <a class="txt" href="#E1:43_16_4_1_2_1"><span class="lab"><font color="Green" title="E98">A353</font></span></a>, <a class="txt" href="#E2:43_16_4_1_2_1"><span class="lab"><font color="Green" title="E99">A354</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E4:43_16_4_1_2_1"/><span class="lab"><font color="Green" title="E101">A356</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix14.html#T23" target="_self" title="MATRIX14:th.23">Th23</a></span>;<br/>
<a NAME="E5:43_16_4_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A357</font></span>: 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_16_4_1_2_1"><span class="lab"><font color="Green" title="E98">A353</font></span></a>, <a class="txt" href="#E2:43_16_4_1_2_1"><span class="lab"><font color="Green" title="E99">A354</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/>
<a NAME="E6:43_16_4_1_2_1"/>
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:43"><span class="lab"><font color="Green" title="E66">A204</font></span></a>, <a class="txt" href="#E1:43_16_4_1_2_1"><span class="lab"><font color="Green" title="E98">A353</font></span></a>, <a class="txt" href="#E2:43_16_4_1_2_1"><span class="lab"><font color="Green" title="E99">A354</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/>
<a NAME="E7:43_16_4_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A358</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43_16_4_1_2_1"><span class="lab"><font color="Green" title="E102">A357</font></span></a>, <a class="ref" href="fvsum_1.html#T62" title="FVSUM_1:th.62">FVSUM_1:62</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_16_4_1_2_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="E1:43_16_4_1_2_1_1"/>
( 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</font> or 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">k</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_16_4_1_2_1"><span class="lab"><font color="Green" title="E98">A353</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_16_4_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:43_16_4_1_2_1_1_1"/><span class="lab"><font color="Green" title="E104">A359</font></span>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">k</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="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/><div class="add"><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A360</font></span>: <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> = 
<font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_16_4_1_2_1"><span class="lab"><font color="Green" title="E99">A354</font></span></a>, <a class="txt" href="#E5:43_16_4_1_2_1"><span class="lab"><font color="Green" title="E102">A357</font></span></a>, <a class="txt" href="#E7:43_16_4_1_2_1"><span class="lab"><font color="Green" title="E103">A358</font></span></a>, <a class="ref" href="matrix14.html#T24" target="_self" title="MATRIX14:th.24">Th24</a></span>
<br/>.= 
<font color="Maroon" title="c3">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#D16" title="VECTSP_1:def.16">VECTSP_1:def 16</a></span>
;<br/><font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> = 
<font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c7">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_16_4_1_2_1"><span class="lab"><font color="Green" title="E100">A355</font></span></a>, <a class="txt" href="#E1:43_16_4_1_2_1_1_1"><span class="lab"><font color="Green" title="E104">A359</font></span></a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></span>
<br/>.= 
<font color="Maroon" title="c3">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43"><span class="lab"><font color="Green" title="E38">A5</font></span></a>, <a class="txt" href="#E4:43_16"><span class="lab"><font color="Green" title="E85">A307</font></span></a>, <a class="txt" href="#E1:43_16_4_1_2"><span class="lab"><font color="Green" title="E96">A351</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="E4:43_16_4_1_2_1_1_1"/>
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_16_4_1_2_1_1_1"><span class="lab"><font color="Green" title="E105">A360</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_16_4_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:43_16_4_1_2_1_1_2"/><span class="lab"><font color="Green" title="E104">A361</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">k</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="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/><div class="add"><a NAME="E2:43_16_4_1_2_1_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c7">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> <font color="Maroon" title="c5">Q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_16"><span class="lab"><font color="Green" title="E85">A307</font></span></a>, <a class="txt" href="#E5:43_16"><span class="lab"><font color="Green" title="E86">A308</font></span></a>, <a class="txt" href="#E1:43_16_4_1_2_1"><span class="lab"><font color="Green" title="E98">A353</font></span></a>, <a class="txt" href="#E2:43_16_4_1_2_1"><span class="lab"><font color="Green" title="E99">A354</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E3:43_16_4_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A362</font></span>: 
 ex <font color="Olive" title="b1">p2</font> being  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font> st <br/>( <font color="Olive" title="b1">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> &amp; <font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</a></span>;<br/><a NAME="E4:43_16_4_1_2_1_1_2"/><span class="lab"><font color="Green" title="E106">A363</font></span>: 
<font color="Maroon" title="c10">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/><span class="lab"><font color="Green" title="E107">A364</font></span>: <font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> = 
<font color="Maroon" title="c5">Q</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">k</font>,<font color="Maroon" title="c7">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_16_4_1_2_1"><span class="lab"><font color="Green" title="E100">A355</font></span></a>, <a class="ref" href="matrix_1.html#D9" title="MATRIX_1:def.9">MATRIX_1:def 9</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c10">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:43"><span class="lab"><font color="Green" title="E40">A7</font></span></a>, <a class="txt" href="#E2:43_16_4_1_2_1"><span class="lab"><font color="Green" title="E99">A354</font></span></a>, <a class="txt" href="#E1:43_16_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E104">A361</font></span></a>, <a class="txt" href="#E3:43_16_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E105">A362</font></span></a>, <a class="txt" href="#E4:43_16_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E106">A363</font></span></a></span>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_16_4_1_2_1_1_2_2"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_16_4_1_2_1_1_2_2_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="E1:43_16_4_1_2_1_1_2_2_1"/>
( <font color="Maroon" title="c10">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">j</font> or <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_16_4_1_2_1_1_2_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:43_16_4_1_2_1_1_2_2_1_1"/><span class="lab"><font color="Green" title="E108">A365</font></span>: 
<font color="Maroon" title="c10">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">j</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="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/><div class="add"><a NAME="E2:43_16_4_1_2_1_1_2_2_1_1"/>
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_16_4_1_2_1"><span class="lab"><font color="Green" title="E98">A353</font></span></a>, <a class="txt" href="#E2:43_16_4_1_2_1"><span class="lab"><font color="Green" title="E99">A354</font></span></a>, <a class="txt" href="#E4:43_16_4_1_2_1"><span class="lab"><font color="Green" title="E101">A356</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">k</font></span>)</span> = 
<font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_16_4_1_2_1"><span class="lab"><font color="Green" title="E99">A354</font></span></a>, <a class="txt" href="#E1:43_16_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E104">A361</font></span></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">Th25</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:43_16_4_1_2_1_1_2_2_1_1"/>
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_16"><span class="lab"><font color="Green" title="E85">A307</font></span></a>, <a class="txt" href="#E5:43_16"><span class="lab"><font color="Green" title="E86">A308</font></span></a>, <a class="txt" href="#E7:43_16_4_1_2_1"><span class="lab"><font color="Green" title="E103">A358</font></span></a>, <a class="txt" href="#E5:43_16_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E107">A364</font></span></a>, <a class="txt" href="#E1:43_16_4_1_2_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E108">A365</font></span></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">Th25</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="43_16_4_1_2_1_1_2_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:43_16_4_1_2_1_1_2_2_1_2"/>
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</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="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:43_16_4_1_2_1_1_2_2_1_2"/>
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_16_4_1_2"><span class="lab"><font color="Green" title="E96">A351</font></span></a>, <a class="txt" href="#E1:43_16_4_1_2_1_1_2"><span class="lab"><font color="Green" title="E104">A361</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E7:43_16_4_1_2_1_1_2"/>
<font color="Maroon" title="c8">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font>
 ; <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 NAME="E4:43_16_4_1_2"/><span class="lab"><font color="Green" title="E98">A366</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_16"><span class="lab"><font color="Green" title="E83">A305</font></span></a>, <a class="txt" href="#E3:43_16"><span class="lab"><font color="Green" title="E84">A306</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E5:43_16_4_1_2"/><span class="lab"><font color="Green" title="E99">A367</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:43"><span class="lab"><font color="Green" title="E72">A223</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A368</font></span>: <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:43"><span class="lab"><font color="Green" title="E68">A219</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>
<br/>.= 
<font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:43"><span class="lab"><font color="Green" title="E75">A226</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>
;<br/><span class="lab"><font color="Green" title="E101">A369</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> = 
<span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">Q</font>,<font color="Maroon" title="c7">j</font></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:43"><span class="lab"><font color="Green" title="E64">A202</font></span></a>, <a class="txt" href="#E42:43"><span class="lab"><font color="Green" title="E72">A223</font></span></a>, <a class="txt" href="#E7:43_16"><span class="lab"><font color="Green" title="E88">A310</font></span></a>, <a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a></span>
<br/>.= 
<span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="fvsum_1.html#K9" title="FVSUM_1:func.9">*</a> <span class="p3">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:43_16"><span class="lab"><font color="Green" title="E90">A312</font></span></a>, <a class="txt" href="#E1:43_16"><span class="lab"><font color="Green" title="E82">A304</font></span></a>, <a class="txt" href="#E3:43_16_4_1_2"><span class="lab"><font color="Green" title="E97">A352</font></span></a>, <a class="ref" href="finseq_1.html#T18" title="FINSEQ_1:th.18">FINSEQ_1:18</a></span>
<br/>.= 
<font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,<font color="Maroon" title="c6">i</font></span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:43"><span class="lab"><font color="Green" title="E65">A203</font></span></a>, <a class="txt" href="#E5:43_16_4_1_2"><span class="lab"><font color="Green" title="E99">A367</font></span></a>, <a class="ref" href="matrix14.html#T10" target="_self" title="MATRIX14:th.10">Th10</a></span>
<br/>.= 
<font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:43"><span class="lab"><font color="Green" title="E68">A219</font></span></a>, <a class="txt" href="#E5:43_16_4_1_2"><span class="lab"><font color="Green" title="E99">A367</font></span></a>, <a class="txt" href="#E6:43_16_4_1_2"><span class="lab"><font color="Green" title="E100">A368</font></span></a>, <a class="ref" href="matrix14.html#T35" target="_self" title="MATRIX14:th.35">Th35</a></span>
;<br/><span class="kw">consider </span><font color="Maroon" title="c10">p2</font> being   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <font color="Maroon" title="c2">K</font><span class="kw"> such that </span><br/><a NAME="E9:43_16_4_1_2"/><span class="lab"><font color="Green" title="E102">A370</font></span>: 
<font color="Maroon" title="c10">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</font>
 <span class="kw">and </span><br/><a NAME="E10:43_16_4_1_2"/><span class="lab"><font color="Green" title="E103">A371</font></span>: 
<font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:43_16"><span class="lab"><font color="Green" title="E91">A313</font></span></a>, <a class="ref" href="matrix_1.html#D6" title="MATRIX_1:def.6">MATRIX_1:def 6</a></span>;<br/><a NAME="E11:43_16_4_1_2"/>
<font color="Maroon" title="c10">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c6">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43"><span class="lab"><font color="Green" title="E37">A4</font></span></a>, <a class="txt" href="#E2:43_16"><span class="lab"><font color="Green" title="E83">A305</font></span></a>, <a class="txt" href="#E3:43_16"><span class="lab"><font color="Green" title="E84">A306</font></span></a>, <a class="txt" href="#E9:43_16_4_1_2"><span class="lab"><font color="Green" title="E102">A370</font></span></a></span>;<br/><span class="kw">hence </span><span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> = 
<font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_16"><span class="lab"><font color="Green" title="E83">A305</font></span></a>, <a class="txt" href="#E4:43_16_4_1_2"><span class="lab"><font color="Green" title="E98">A366</font></span></a>, <a class="txt" href="#E7:43_16_4_1_2"><span class="lab"><font color="Green" title="E101">A369</font></span></a>, <a class="txt" href="#E10:43_16_4_1_2"><span class="lab"><font color="Green" title="E103">A371</font></span></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">Th25</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></span>
;<br/> <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>
<span class="kw">hence </span><a NAME="E16:43_16"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="lab"><font color="Green" title="E82">A372</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,1</span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrixr1.html#T16" title="MATRIXR1:th.16">MATRIXR1:16</a></span>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:43"><span class="lab"><font color="Green" title="E72">A223</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>
;<br/>
<a NAME="E54:43"/>
(  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <font color="Maroon" title="c4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<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>,<span class="p2">(<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></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default">1,1</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> <font color="Maroon" title="c4">P</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:43"><span class="lab"><font color="Green" title="E68">A219</font></span></a>, <a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E83">A373</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1 = 
<span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K9" title="MATRIX_1:func.9">Col</a> <font color="Maroon" title="c5">Q</font>,1</span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:43"><span class="lab"><font color="Green" title="E64">A202</font></span></a>, <a class="txt" href="#E42:43"><span class="lab"><font color="Green" title="E72">A223</font></span></a>, <a class="txt" href="#E51:43"><span class="lab"><font color="Green" title="E80">A302</font></span></a>, <a class="ref" href="matrix_3.html#D4" title="MATRIX_3:def.4">MATRIX_3:def 4</a></span>
<br/>.= 
<font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1"><a href="matrix14.html#NK3" title="MATRIX14:NK.3">|(</a><span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,1</span>)</span></span><a href="matrix14.html#NK3" title="MATRIX14:NK.3">)|</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:43"><span class="lab"><font color="Green" title="E73">A224</font></span></a>, <a class="txt" href="#E35:43"><span class="lab"><font color="Green" title="E65">A203</font></span></a>, <a class="txt" href="#E50:43"><span class="lab"><font color="Green" title="E79">A301</font></span></a>, <a class="ref" href="matrix14.html#T10" target="_self" title="MATRIX14:th.10">Th10</a></span>
<br/>.= 
<font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:43"><span class="lab"><font color="Green" title="E73">A224</font></span></a>, <a class="txt" href="#E38:43"><span class="lab"><font color="Green" title="E68">A219</font></span></a>, <a class="ref" href="matrix14.html#T35" target="_self" title="MATRIX14:th.35">Th35</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:43"><span class="lab"><font color="Green" title="E74">A225</font></span></a>, <a class="ref" href="polynom1.html#D2" title="POLYNOM1:def.2">POLYNOM1:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="fvsum_1.html#K10" title="FVSUM_1:func.10">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c4">P</font>,1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:43"><span class="lab"><font color="Green" title="E68">A219</font></span></a>, <a class="txt" href="#E53:43"><span class="lab"><font color="Green" title="E82">A372</font></span></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></span>
<br/>.= 
<font color="Maroon" title="c3">a</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:43"><span class="lab"><font color="Green" title="E75">A226</font></span></a>, <a class="txt" href="#E41:43"><span class="lab"><font color="Green" title="E71">A222</font></span></a>, <a class="ref" href="matrix12.html#T3" title="MATRIX12:th.3">MATRIX12:3</a></span>
<br/>.= 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E36">A3</font></span></a>, <a class="ref" href="vectsp_1.html#D22" title="VECTSP_1:def.22">VECTSP_1:def 22</a></span>
;<br/>
<a NAME="E56:43"/>
for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">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="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_19"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">i</font>, <font color="Maroon" title="c7">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="c6">i</font>,<font color="Maroon" title="c7">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="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c8">i0</font> = <font color="Maroon" title="c6">i</font>, <font color="Maroon" title="c9">j0</font> = <font color="Maroon" title="c7">j</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <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/>
<span class="kw">assume </span><a NAME="E2:43_19"/><span class="lab"><font color="Green" title="E84">A374</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">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="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></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"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span><br/>

<a NAME="E3:43_19"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A375</font></span>: 
<font color="Maroon" title="c6">i</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> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E51:43"><span class="lab"><font color="Green" title="E80">A302</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E4:43_19"/><span class="kw">then </span><span class="lab"><font color="Green" title="E86">A376</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E5:43_19"/><span class="lab"><font color="Green" title="E87">A377</font></span>: 
<font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_19"><span class="lab"><font color="Green" title="E85">A375</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E6:43_19"/><span class="lab"><font color="Green" title="E88">A378</font></span>: 
<font color="Maroon" title="c7">j</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> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E51:43"><span class="lab"><font color="Green" title="E80">A302</font></span></a>, <a class="txt" href="#E2:43_19"><span class="lab"><font color="Green" title="E84">A374</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E7:43_19"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A379</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E8:43_19"/><span class="lab"><font color="Green" title="E90">A380</font></span>: 
<font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:43_19"><span class="lab"><font color="Green" title="E88">A378</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_19_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="E1:43_19_1"/>
( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">i</font> or 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_19"><span class="lab"><font color="Green" title="E86">A376</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_19_1_1"><span class="kw">suppose </span></a><a NAME="E1:43_19_1_1"/><span class="lab"><font color="Green" title="E91">A381</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">i</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="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_19_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_19_1_1_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="E1:43_19_1_1_1_1"/>
( <font color="Maroon" title="c6">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">j</font> or <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_19_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:43_19_1_1_1_1_1"/><span class="lab"><font color="Green" title="E92">A382</font></span>: 
<font color="Maroon" title="c6">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">j</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="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span><br/><div class="add"><span class="lab"><font color="Green" title="E93">A383</font></span>: <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> = 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c8">i0</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_19"><span class="lab"><font color="Green" title="E86">A376</font></span></a>, <a class="txt" href="#E5:43_19"><span class="lab"><font color="Green" title="E87">A377</font></span></a>, <a class="txt" href="#E7:43_19"><span class="lab"><font color="Green" title="E89">A379</font></span></a>, <a class="txt" href="#E8:43_19"><span class="lab"><font color="Green" title="E90">A380</font></span></a>, <a class="ref" href="matrix14.html#T27" target="_self" title="MATRIX14:th.27">Th27</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:43_19"><span class="lab"><font color="Green" title="E89">A379</font></span></a>, <a class="txt" href="#E8:43_19"><span class="lab"><font color="Green" title="E90">A380</font></span></a>, <a class="txt" href="#E1:43_19_1_1_1_1_1"><span class="lab"><font color="Green" title="E92">A382</font></span></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">Th25</a></span>
;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">i0</font>,<font color="Maroon" title="c9">j0</font>

<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E52:43"><span class="lab"><font color="Green" title="E81">A303</font></span></a>, <a class="txt" href="#E5:43_19"><span class="lab"><font color="Green" title="E87">A377</font></span></a>, <a class="txt" href="#E7:43_19"><span class="lab"><font color="Green" title="E89">A379</font></span></a>, <a class="txt" href="#E8:43_19"><span class="lab"><font color="Green" title="E90">A380</font></span></a>, <a class="txt" href="#E1:43_19_1_1"><span class="lab"><font color="Green" title="E91">A381</font></span></a>, <a class="txt" href="#E1:43_19_1_1_1_1_1"><span class="lab"><font color="Green" title="E92">A382</font></span></a>, <a class="txt" href="#E2:43_19_1_1_1_1_1"><span class="lab"><font color="Green" title="E93">A383</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_19_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:43_19_1_1_1_1_2"/><span class="lab"><font color="Green" title="E92">A384</font></span>: 
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</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="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span><br/><div class="add"><span class="lab"><font color="Green" title="E93">A385</font></span>: <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> = 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c8">i0</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_19"><span class="lab"><font color="Green" title="E86">A376</font></span></a>, <a class="txt" href="#E5:43_19"><span class="lab"><font color="Green" title="E87">A377</font></span></a>, <a class="txt" href="#E7:43_19"><span class="lab"><font color="Green" title="E89">A379</font></span></a>, <a class="txt" href="#E8:43_19"><span class="lab"><font color="Green" title="E90">A380</font></span></a>, <a class="ref" href="matrix14.html#T27" target="_self" title="MATRIX14:th.27">Th27</a></span>
<br/>.= 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:43_19"><span class="lab"><font color="Green" title="E89">A379</font></span></a>, <a class="txt" href="#E8:43_19"><span class="lab"><font color="Green" title="E90">A380</font></span></a>, <a class="txt" href="#E1:43_19_1_1_1_1_2"><span class="lab"><font color="Green" title="E92">A384</font></span></a>, <a class="ref" href="matrix14.html#T24" target="_self" title="MATRIX14:th.24">Th24</a></span>
;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">i0</font>,<font color="Maroon" title="c9">j0</font>

<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:43"><span class="lab"><font color="Green" title="E78">A258</font></span></a>, <a class="txt" href="#E8:43_19"><span class="lab"><font color="Green" title="E90">A380</font></span></a>, <a class="txt" href="#E1:43_19_1_1"><span class="lab"><font color="Green" title="E91">A381</font></span></a>, <a class="txt" href="#E1:43_19_1_1_1_1_2"><span class="lab"><font color="Green" title="E92">A384</font></span></a>, <a class="txt" href="#E2:43_19_1_1_1_1_2"><span class="lab"><font color="Green" title="E93">A385</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:43_19_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font>
 ; <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="43_19_1_2"><span class="kw">suppose </span></a><a NAME="E1:43_19_1_2"/><span class="lab"><font color="Green" title="E91">A386</font></span>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i</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="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_19_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_19_1_2_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="E1:43_19_1_2_1_1"/>
( <font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j</font> or <font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_19_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:43_19_1_2_1_1_1"/><span class="lab"><font color="Green" title="E92">A387</font></span>: 
<font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">j</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="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span><br/><div class="add"><span class="lab"><font color="Green" title="E93">A388</font></span>: <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> = 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c8">i0</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_19"><span class="lab"><font color="Green" title="E86">A376</font></span></a>, <a class="txt" href="#E5:43_19"><span class="lab"><font color="Green" title="E87">A377</font></span></a>, <a class="txt" href="#E7:43_19"><span class="lab"><font color="Green" title="E89">A379</font></span></a>, <a class="txt" href="#E8:43_19"><span class="lab"><font color="Green" title="E90">A380</font></span></a>, <a class="ref" href="matrix14.html#T27" target="_self" title="MATRIX14:th.27">Th27</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">K</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:43_19"><span class="lab"><font color="Green" title="E89">A379</font></span></a>, <a class="txt" href="#E8:43_19"><span class="lab"><font color="Green" title="E90">A380</font></span></a>, <a class="txt" href="#E1:43_19_1_2_1_1_1"><span class="lab"><font color="Green" title="E92">A387</font></span></a>, <a class="ref" href="matrix14.html#T25" target="_self" title="MATRIX14:th.25">Th25</a></span>
;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">i0</font>,<font color="Maroon" title="c9">j0</font>

<br/>.= 
<span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:43"><span class="lab"><font color="Green" title="E76">A227</font></span></a>, <a class="txt" href="#E8:43_19"><span class="lab"><font color="Green" title="E90">A380</font></span></a>, <a class="txt" href="#E1:43_19_1_2"><span class="lab"><font color="Green" title="E91">A386</font></span></a>, <a class="txt" href="#E1:43_19_1_2_1_1_1"><span class="lab"><font color="Green" title="E92">A387</font></span></a>, <a class="txt" href="#E2:43_19_1_2_1_1_1"><span class="lab"><font color="Green" title="E93">A388</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_19_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:43_19_1_2_1_1_2"/><span class="lab"><font color="Green" title="E92">A389</font></span>: 
<font color="Maroon" title="c6">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c7">j</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="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font></span><br/><div class="add"><a NAME="E2:43_19_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A390</font></span>: 
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:43_19"><span class="lab"><font color="Green" title="E89">A379</font></span></a>, <a class="txt" href="#E1:43_19_1_2"><span class="lab"><font color="Green" title="E91">A386</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> = 
<span class="p1">(<span class="default"><a href="matrix14.html#K4" title="MATRIX14:func.4">Base_FinSeq</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>,<font color="Maroon" title="c8">i0</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">j0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_19"><span class="lab"><font color="Green" title="E86">A376</font></span></a>, <a class="txt" href="#E5:43_19"><span class="lab"><font color="Green" title="E87">A377</font></span></a>, <a class="txt" href="#E7:43_19"><span class="lab"><font color="Green" title="E89">A379</font></span></a>, <a class="txt" href="#E8:43_19"><span class="lab"><font color="Green" title="E90">A380</font></span></a>, <a class="ref" href="matrix14.html#T27" target="_self" title="MATRIX14:th.27">Th27</a></span>
<br/>.= 
 <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c2">K</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:43_19"><span class="lab"><font color="Green" title="E89">A379</font></span></a>, <a class="txt" href="#E8:43_19"><span class="lab"><font color="Green" title="E90">A380</font></span></a>, <a class="txt" href="#E2:43_19_1_2_1_1_2"><span class="lab"><font color="Green" title="E93">A390</font></span></a>, <a class="ref" href="matrix14.html#T24" target="_self" title="MATRIX14:th.24">Th24</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:43_19_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E55:43"><span class="lab"><font color="Green" title="E83">A373</font></span></a>, <a class="txt" href="#E7:43_19"><span class="lab"><font color="Green" title="E89">A379</font></span></a>, <a class="txt" href="#E1:43_19_1_2"><span class="lab"><font color="Green" title="E91">A386</font></span></a>, <a class="txt" href="#E1:43_19_1_2_1_1_2"><span class="lab"><font color="Green" title="E92">A389</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</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><span class="kw">hence </span><a NAME="E3:43_19_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i</font>,<font color="Maroon" title="c7">j</font>
 ; <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 NAME="E57:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A391</font></span>: 
<font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K4" title="MATRIX_6:func.4">*</a> <font color="Maroon" title="c5">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K12" title="MATRIX_1:func.12">1.</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T28" title="MATRIX_1:th.28">MATRIX_1:28</a></span>;<br/>
<span class="kw">hence </span><a NAME="E58:43"/>
<font color="Maroon" title="c4">P</font> is  <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:43"><span class="lab"><font color="Green" title="E63">A201</font></span></a>, <a class="ref" href="matrix14.html#T19" target="_self" title="MATRIX14:th.19">Th19</a></span>; <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="c5">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a> </span><br/>

<span class="kw">thus </span><a NAME="E59:43"/>
<font color="Maroon" title="c5">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">P</font> <a href="matrix_6.html#K5" title="MATRIX_6:func.5">~</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:43"><span class="lab"><font color="Green" title="E84">A391</font></span></a>, <a class="txt" href="#E33:43"><span class="lab"><font color="Green" title="E63">A201</font></span></a>, <a class="ref" href="matrix14.html#T18" target="_self" title="MATRIX14:th.18">Th18</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>
