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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">K</font> being  <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a><br/> for <font color="Olive" title="b2">M</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><br/> for <font color="Olive" title="b3">i</font>, <font color="Olive" title="b4">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">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> &amp; <font color="Olive" title="b4">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> holds <br/>for <font color="Olive" title="b5">k</font>, <font color="Olive" title="b6">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b5">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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> &amp; <font color="Olive" title="b6">m</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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> holds <br/>( ( <font color="Olive" title="b5">k</font> <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="b6">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">j</font> implies <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Olive" title="b2">M</font>,<font color="Olive" title="b3">i</font>,<font color="Olive" title="b4">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b5">k</font>,<font color="Olive" title="b6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b5">k</font>,<font color="Olive" title="b6">m</font> ) &amp; ( <font color="Olive" title="b5">k</font> <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="b6">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b4">j</font> implies <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Olive" title="b2">M</font>,<font color="Olive" title="b3">i</font>,<font color="Olive" title="b4">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b5">k</font>,<font color="Olive" title="b6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b5">k</font>,<span class="p1">(<span class="default"><font color="Olive" title="b6">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> ) &amp; ( <font color="Olive" title="b5">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b6">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">j</font> implies <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Olive" title="b2">M</font>,<font color="Olive" title="b3">i</font>,<font color="Olive" title="b4">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b5">k</font>,<font color="Olive" title="b6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Olive" title="b6">m</font> ) &amp; ( <font color="Olive" title="b5">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b3">i</font> &amp; <font color="Olive" title="b6">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b4">j</font> implies <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Olive" title="b2">M</font>,<font color="Olive" title="b3">i</font>,<font color="Olive" title="b4">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b5">k</font>,<font color="Olive" title="b6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b6">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">K</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">M</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><br/> for <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">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> &amp; <font color="Olive" title="b3">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> holds <br/>for <font color="Olive" title="b4">k</font>, <font color="Olive" title="b5">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b4">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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> &amp; <font color="Olive" title="b5">m</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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> holds <br/>( ( <font color="Olive" title="b4">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b5">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">j</font> implies <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Olive" title="b1">M</font>,<font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b4">k</font>,<font color="Olive" title="b5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b4">k</font>,<font color="Olive" title="b5">m</font> ) &amp; ( <font color="Olive" title="b4">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b5">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b3">j</font> implies <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Olive" title="b1">M</font>,<font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b4">k</font>,<font color="Olive" title="b5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b4">k</font>,<span class="p1">(<span class="default"><font color="Olive" title="b5">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> ) &amp; ( <font color="Olive" title="b4">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b5">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">j</font> implies <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Olive" title="b1">M</font>,<font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b4">k</font>,<font color="Olive" title="b5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Olive" title="b5">m</font> ) &amp; ( <font color="Olive" title="b4">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b5">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b3">j</font> implies <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Olive" title="b1">M</font>,<font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b4">k</font>,<font color="Olive" title="b5">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b4">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b5">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">M</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"> 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 <font color="Olive" title="b1">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> &amp; <font color="Olive" title="b2">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> holds <br/>for <font color="Olive" title="b3">k</font>, <font color="Olive" title="b4">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b3">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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> &amp; <font color="Olive" title="b4">m</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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> holds <br/>( ( <font color="Olive" title="b3">k</font> <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="b4">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">j</font> implies <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b3">k</font>,<font color="Olive" title="b4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b3">k</font>,<font color="Olive" title="b4">m</font> ) &amp; ( <font color="Olive" title="b3">k</font> <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="b4">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b2">j</font> implies <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b3">k</font>,<font color="Olive" title="b4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b3">k</font>,<span class="p1">(<span class="default"><font color="Olive" title="b4">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> ) &amp; ( <font color="Olive" title="b3">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b4">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">j</font> implies <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b3">k</font>,<font color="Olive" title="b4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Olive" title="b4">m</font> ) &amp; ( <font color="Olive" title="b3">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b4">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Olive" title="b2">j</font> implies <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b3">k</font>,<font color="Olive" title="b4">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b4">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">i</font>, <font color="Maroon" title="c5">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"> ( <font color="Maroon" title="c4">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> &amp; <font color="Maroon" title="c5">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> implies for <font color="Olive" title="b1">k</font>, <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> &amp; <font color="Olive" title="b2">m</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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> holds <br/>( ( <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">j</font> implies <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">m</font> ) &amp; ( <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">j</font> implies <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">k</font>,<span class="p1">(<span class="default"><font color="Olive" title="b2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> ) &amp; ( <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">j</font> implies <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Olive" title="b2">m</font> ) &amp; ( <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">j</font> implies <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> ) ) )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:15"/><span class="lab"><font color="Green" title="E15">A1</font></span>: 
<font color="Maroon" title="c4">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">and </span><br/><a NAME="E2:15"/><span class="lab"><font color="Green" title="E16">A2</font></span>: 
<font color="Maroon" title="c5">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>
 ; <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>, <font color="Olive" title="b2">m</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> &amp; <font color="Olive" title="b2">m</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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> holds <br/>( ( <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">j</font> implies <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">m</font> ) &amp; ( <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">j</font> implies <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">k</font>,<span class="p1">(<span class="default"><font color="Olive" title="b2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> ) &amp; ( <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">j</font> implies <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Olive" title="b2">m</font> ) &amp; ( <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">j</font> implies <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b2">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> ) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c6">DM</font> =  <a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font>;<br/>
<a NAME="E3:15"/><span class="lab"><font color="Green" title="E17">A3</font></span>: 
 <a href="matrix_2.html#K9" title="MATRIX_2:func.9">Deleting</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="txt" href="#E2:15"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="ref" href="laplace.html#D1" target="_self" title="LAPLACE:def.1">Def1</a></span>;<br/>
<a NAME="E4:15"/>
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15"><span class="lab"><font color="Green" title="E15">A1</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">n9</font> = <font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 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="nat_1.html#T20" title="NAT_1:th.20">NAT_1:20</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">DL</font> =  <a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c9">k</font>, <font color="Maroon" title="c10">m</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"> ( <font color="Maroon" title="c9">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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> &amp; <font color="Maroon" title="c10">m</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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> implies ( ( <font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Maroon" title="c10">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">j</font> implies <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c10">m</font> ) &amp; ( <font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Maroon" title="c10">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">j</font> implies <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">k</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> ) &amp; ( <font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Maroon" title="c10">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">j</font> implies <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c10">m</font> ) &amp; ( <font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Maroon" title="c10">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">j</font> implies <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> ) ) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E6:15"/><span class="lab"><font color="Green" title="E18">A4</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#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E7:15"/><span class="lab"><font color="Green" title="E19">A5</font></span>: 
<font color="Maroon" title="c10">m</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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</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="c9">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Maroon" title="c10">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">j</font> implies <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c10">m</font> ) &amp; ( <font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Maroon" title="c10">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">j</font> implies <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">k</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> ) &amp; ( <font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Maroon" title="c10">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">j</font> implies <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c10">m</font> ) &amp; ( <font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Maroon" title="c10">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">j</font> implies <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> ) )</span><br/>

<a NAME="E8:15"/><span class="lab"><font color="Green" title="E20">A6</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>;<br/>
<a NAME="E9:15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A7</font></span>: 
<font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 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"><font color="Maroon" title="c7">n9</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:15"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="ref" href="finseq_1.html#T81" title="FINSEQ_1:th.81">FINSEQ_1:81</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">I</font> = <font color="Maroon" title="c4">i</font>, <font color="Maroon" title="c12">J</font> = <font color="Maroon" title="c5">j</font>, <font color="Maroon" title="c13">K</font> = <font color="Maroon" title="c9">k</font>, <font color="Maroon" title="c14">U</font> = <font color="Maroon" title="c10">m</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/>
<a NAME="E11:15"/>
<font color="Maroon" title="c7">n9</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">n9</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<a NAME="E12:15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A8</font></span>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">n9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T7" title="FINSEQ_1:th.7">FINSEQ_1:7</a></span>;<br/>
<a NAME="E13:15"/><span class="lab"><font color="Green" title="E23">A9</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">M</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="E14:15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A10</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font>
 
<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="E15:15"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="txt" href="#E8:15"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="txt" href="#E13:15"><span class="lab"><font color="Green" title="E23">A9</font></span></a>, <a class="ref" href="laplace.html#T1" target="_self" title="LAPLACE:th.1">Th1</a></span>;<br/>
<a NAME="E16:15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A11</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">n9</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="E17:15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A12</font></span>: 
<span class="p1">(<span class="default"><a href="matrix_2.html#K9" title="MATRIX_2:func.9">Deleting</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <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="finseq_3.html#K2" title="FINSEQ_3:func.2">Del</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c9">k</font></span>)</span>,<font color="Maroon" title="c5">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:15"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="txt" href="#E8:15"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="ref" href="matrix_2.html#D6" title="MATRIX_2:def.6">MATRIX_2:def 6</a></span>;<br/>
<a NAME="E18:15"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:15"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></span>;<br/>
<a NAME="E19:15"/><span class="kw">then </span>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c7">n9</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="E20:15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A13</font></span>: 
<span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <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="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span>,<font color="Maroon" title="c9">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:15"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="txt" href="#E8:15"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="ref" href="matrix_2.html#T18" title="MATRIX_2:th.18">MATRIX_2:18</a></span>;<br/>
<a NAME="E21:15"/>
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:15"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></span>;<br/>
<a NAME="E22:15"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A14</font></span>: 
<span class="p1">(<span class="default"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span>,<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="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c10">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:15"><span class="lab"><font color="Green" title="E19">A5</font></span></a>, <a class="txt" href="#E8:15"><span class="lab"><font color="Green" title="E20">A6</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="E23:15"/><span class="lab"><font color="Green" title="E29">A15</font></span>: 
 <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p1">(<span class="default"><a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font></span>)</span>,<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="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</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="#E6:15"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="txt" href="#E8:15"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="txt" href="#E16:15"><span class="lab"><font color="Green" title="E25">A11</font></span></a>, <a class="ref" href="matrix_2.html#T18" title="MATRIX_2:th.18">MATRIX_2:18</a></span>;<br/>
<a NAME="E24:15"/><span class="lab"><font color="Green" title="E30">A16</font></span>: 
<font color="Maroon" title="c10">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 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"><font color="Maroon" title="c7">n9</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:15"><span class="lab"><font color="Green" title="E19">A5</font></span></a>, <a class="txt" href="#E8:15"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="ref" href="finseq_1.html#T81" title="FINSEQ_1:th.81">FINSEQ_1:81</a></span>;<br/>
<a NAME="E25:15"/><span class="lab"><font color="Green" title="E31">A17</font></span>: 
( <font color="Maroon" title="c13">K</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c11">I</font> implies ( ( <font color="Maroon" title="c14">U</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">J</font> implies <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">K</font>,<font color="Maroon" title="c14">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">K</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">U</font> ) &amp; ( <font color="Maroon" title="c14">U</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c12">J</font> implies <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">K</font>,<font color="Maroon" title="c14">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">K</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">U</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:15_1"/><span class="lab"><font color="Green" title="E32">A18</font></span>: 
<font color="Maroon" title="c13">K</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c11">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="c14">U</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">J</font> implies <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">K</font>,<font color="Maroon" title="c14">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">K</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">U</font> ) &amp; ( <font color="Maroon" title="c14">U</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c12">J</font> implies <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">K</font>,<font color="Maroon" title="c14">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">K</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">U</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) )</span><br/>

<a NAME="E2:15_1"/>
<font color="Maroon" title="c13">K</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">n9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:15"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="txt" href="#E8:15"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E3:15_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A19</font></span>: 
<span class="p1">(<span class="default"><a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">K</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:15"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="txt" href="#E13:15"><span class="lab"><font color="Green" title="E23">A9</font></span></a>, <a class="txt" href="#E14:15"><span class="lab"><font color="Green" title="E24">A10</font></span></a>, <a class="txt" href="#E9:15"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="txt" href="#E1:15_1"><span class="lab"><font color="Green" title="E32">A18</font></span></a>, <a class="ref" href="finseq_3.html#T120" title="FINSEQ_3:th.120">FINSEQ_3:120</a></span>;<br/>
<a NAME="E4:15_1"/><span class="lab"><font color="Green" title="E34">A20</font></span>: 
<font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">K</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">M</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c13">K</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:15"><span class="lab"><font color="Green" title="E24">A10</font></span></a>, <a class="txt" href="#E9:15"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="ref" href="matrix_2.html#T18" title="MATRIX_2:th.18">MATRIX_2:18</a></span>;<br/>
<span class="kw">thus </span><a NAME="E5:15_1"/>
( <font color="Maroon" title="c14">U</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">J</font> implies <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">K</font>,<font color="Maroon" title="c14">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">K</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">U</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="c14">U</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c12">J</font> implies <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">K</font>,<font color="Maroon" title="c14">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">K</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">U</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:15_1_1"/><span class="lab"><font color="Green" title="E35">A21</font></span>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">M</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="kw">assume </span><a NAME="E2:15_1_1"/><span class="lab"><font color="Green" title="E36">A22</font></span>: 
<font color="Maroon" title="c14">U</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">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"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">K</font>,<font color="Maroon" title="c14">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">K</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">U</font></span><br/>

<a NAME="E3:15_1_1"/>
<span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">K</font>,<font color="Maroon" title="c14">U</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="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">K</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">U</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:15"><span class="lab"><font color="Green" title="E26">A12</font></span></a>, <a class="txt" href="#E3:15"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E20:15"><span class="lab"><font color="Green" title="E27">A13</font></span></a>, <a class="txt" href="#E22:15"><span class="lab"><font color="Green" title="E28">A14</font></span></a>, <a class="txt" href="#E23:15"><span class="lab"><font color="Green" title="E29">A15</font></span></a>, <a class="txt" href="#E3:15_1"><span class="lab"><font color="Green" title="E33">A19</font></span></a>, <a class="txt" href="#E4:15_1"><span class="lab"><font color="Green" title="E34">A20</font></span></a>, <a class="txt" href="#E2:15_1_1"><span class="lab"><font color="Green" title="E36">A22</font></span></a>, <a class="ref" href="finseq_3.html#T119" title="FINSEQ_3:th.119">FINSEQ_3:119</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:15_1_1"/>
<span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">K</font>,<font color="Maroon" title="c14">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">K</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c14">U</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:15"><span class="lab"><font color="Green" title="E19">A5</font></span></a>, <a class="txt" href="#E8:15"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="txt" href="#E12:15"><span class="lab"><font color="Green" title="E22">A8</font></span></a>, <a class="txt" href="#E1:15_1_1"><span class="lab"><font color="Green" title="E35">A21</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E6:15_1"/><span class="lab"><font color="Green" title="E35">A23</font></span>: 
<font color="Maroon" title="c14">U</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c12">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"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">K</font>,<font color="Maroon" title="c14">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">K</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">U</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><br/>

<a NAME="E7:15_1"/><span class="lab"><font color="Green" title="E36">A24</font></span>: 
<font color="Maroon" title="c14">U</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">n9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:15"><span class="lab"><font color="Green" title="E19">A5</font></span></a>, <a class="txt" href="#E8:15"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E8:15_1"/><span class="lab"><font color="Green" title="E37">A25</font></span>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">M</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="E9:15_1"/><span class="lab"><font color="Green" title="E38">A26</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> <span class="p2">(<span class="default"><a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c13">K</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:15"><span class="lab"><font color="Green" title="E29">A15</font></span></a>, <a class="txt" href="#E3:15_1"><span class="lab"><font color="Green" title="E33">A19</font></span></a>, <a class="txt" href="#E4:15_1"><span class="lab"><font color="Green" title="E34">A20</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="E10:15_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">J</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"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c13">K</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="txt" href="#E8:15_1"><span class="lab"><font color="Green" title="E37">A25</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="E11:15_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">K</font>,<font color="Maroon" title="c14">U</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="c3">M</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">K</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">U</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:15"><span class="lab"><font color="Green" title="E26">A12</font></span></a>, <a class="txt" href="#E3:15"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E20:15"><span class="lab"><font color="Green" title="E27">A13</font></span></a>, <a class="txt" href="#E22:15"><span class="lab"><font color="Green" title="E28">A14</font></span></a>, <a class="txt" href="#E23:15"><span class="lab"><font color="Green" title="E29">A15</font></span></a>, <a class="txt" href="#E9:15"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="txt" href="#E3:15_1"><span class="lab"><font color="Green" title="E33">A19</font></span></a>, <a class="txt" href="#E4:15_1"><span class="lab"><font color="Green" title="E34">A20</font></span></a>, <a class="txt" href="#E6:15_1"><span class="lab"><font color="Green" title="E35">A23</font></span></a>, <a class="txt" href="#E9:15_1"><span class="lab"><font color="Green" title="E38">A26</font></span></a>, <a class="txt" href="#E7:15_1"><span class="lab"><font color="Green" title="E36">A24</font></span></a>, <a class="ref" href="finseq_3.html#T120" title="FINSEQ_3:th.120">FINSEQ_3:120</a>, <a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:15_1"/>
<span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">K</font>,<font color="Maroon" title="c14">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">K</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">U</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:15"><span class="lab"><font color="Green" title="E30">A16</font></span></a>, <a class="txt" href="#E8:15_1"><span class="lab"><font color="Green" title="E37">A25</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</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="E26:15"/>
( <font color="Maroon" title="c13">K</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">I</font> implies ( ( <font color="Maroon" title="c14">U</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">J</font> implies <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">K</font>,<font color="Maroon" title="c14">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">K</font>,<font color="Maroon" title="c14">U</font> ) &amp; ( <font color="Maroon" title="c14">U</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c12">J</font> implies <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">K</font>,<font color="Maroon" title="c14">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">K</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">U</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:15_2"/>
<font color="Maroon" title="c13">K</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c11">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="c14">U</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">J</font> implies <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">K</font>,<font color="Maroon" title="c14">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">K</font>,<font color="Maroon" title="c14">U</font> ) &amp; ( <font color="Maroon" title="c14">U</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c12">J</font> implies <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">K</font>,<font color="Maroon" title="c14">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">K</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">U</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) )</span><br/>

<a NAME="E2:15_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A27</font></span>: 
<span class="p1">(<span class="default"><a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T119" title="FINSEQ_3:th.119">FINSEQ_3:119</a></span>;<br/>
<a NAME="E3:15_2"/><span class="lab"><font color="Green" title="E33">A28</font></span>: 
<font color="Maroon" title="c3">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">K</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c13">K</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:15"><span class="lab"><font color="Green" title="E18">A4</font></span></a>, <a class="txt" href="#E8:15"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="txt" href="#E14:15"><span class="lab"><font color="Green" title="E24">A10</font></span></a>, <a class="txt" href="#E12:15"><span class="lab"><font color="Green" title="E22">A8</font></span></a>, <a class="ref" href="matrix_2.html#T18" title="MATRIX_2:th.18">MATRIX_2:18</a></span>;<br/>
<span class="kw">thus </span><a NAME="E4:15_2"/>
( <font color="Maroon" title="c14">U</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">J</font> implies <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">K</font>,<font color="Maroon" title="c14">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">K</font>,<font color="Maroon" title="c14">U</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="c14">U</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c12">J</font> implies <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">K</font>,<font color="Maroon" title="c14">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">K</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">U</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:15_2_1"/><span class="lab"><font color="Green" title="E34">A29</font></span>: 
<font color="Maroon" title="c14">U</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c12">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"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">K</font>,<font color="Maroon" title="c14">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">K</font>,<font color="Maroon" title="c14">U</font></span><br/>

<a NAME="E2:15_2_1"/><span class="lab"><font color="Green" title="E35">A30</font></span>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n9</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<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="E3:15_2_1"/>
<span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">K</font>,<font color="Maroon" title="c14">U</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="c3">M</font>,<font color="Maroon" title="c13">K</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">U</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:15"><span class="lab"><font color="Green" title="E26">A12</font></span></a>, <a class="txt" href="#E3:15"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E20:15"><span class="lab"><font color="Green" title="E27">A13</font></span></a>, <a class="txt" href="#E22:15"><span class="lab"><font color="Green" title="E28">A14</font></span></a>, <a class="txt" href="#E23:15"><span class="lab"><font color="Green" title="E29">A15</font></span></a>, <a class="txt" href="#E2:15_2"><span class="lab"><font color="Green" title="E32">A27</font></span></a>, <a class="txt" href="#E3:15_2"><span class="lab"><font color="Green" title="E33">A28</font></span></a>, <a class="txt" href="#E1:15_2_1"><span class="lab"><font color="Green" title="E34">A29</font></span></a>, <a class="ref" href="finseq_3.html#T119" title="FINSEQ_3:th.119">FINSEQ_3:119</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:15_2_1"/>
<span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">K</font>,<font color="Maroon" title="c14">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">K</font>,<font color="Maroon" title="c14">U</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:15"><span class="lab"><font color="Green" title="E19">A5</font></span></a>, <a class="txt" href="#E8:15"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="txt" href="#E12:15"><span class="lab"><font color="Green" title="E22">A8</font></span></a>, <a class="txt" href="#E2:15_2_1"><span class="lab"><font color="Green" title="E35">A30</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E5:15_2"/><span class="lab"><font color="Green" title="E34">A31</font></span>: 
<font color="Maroon" title="c14">U</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c12">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"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">K</font>,<font color="Maroon" title="c14">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">K</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">U</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><br/>

<a NAME="E6:15_2"/><span class="lab"><font color="Green" title="E35">A32</font></span>: 
<font color="Maroon" title="c14">U</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">n9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:15"><span class="lab"><font color="Green" title="E19">A5</font></span></a>, <a class="txt" href="#E8:15"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E7:15_2"/><span class="lab"><font color="Green" title="E36">A33</font></span>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">M</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="E8:15_2"/><span class="lab"><font color="Green" title="E37">A34</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> <span class="p2">(<span class="default"><a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c13">K</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:15"><span class="lab"><font color="Green" title="E29">A15</font></span></a>, <a class="txt" href="#E2:15_2"><span class="lab"><font color="Green" title="E32">A27</font></span></a>, <a class="txt" href="#E3:15_2"><span class="lab"><font color="Green" title="E33">A28</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="E9:15_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">J</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"><a href="matrix_1.html#K8" title="MATRIX_1:func.8">Line</a> <span class="p2">(<span class="default"><a href="matrix_2.html#K8" title="MATRIX_2:func.8">DelLine</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font></span>)</span>,<font color="Maroon" title="c13">K</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:15"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="txt" href="#E7:15_2"><span class="lab"><font color="Green" title="E36">A33</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="E10:15_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">K</font>,<font color="Maroon" title="c14">U</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="c3">M</font>,<font color="Maroon" title="c13">K</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">U</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:15"><span class="lab"><font color="Green" title="E26">A12</font></span></a>, <a class="txt" href="#E3:15"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="txt" href="#E20:15"><span class="lab"><font color="Green" title="E27">A13</font></span></a>, <a class="txt" href="#E22:15"><span class="lab"><font color="Green" title="E28">A14</font></span></a>, <a class="txt" href="#E23:15"><span class="lab"><font color="Green" title="E29">A15</font></span></a>, <a class="txt" href="#E9:15"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="txt" href="#E2:15_2"><span class="lab"><font color="Green" title="E32">A27</font></span></a>, <a class="txt" href="#E3:15_2"><span class="lab"><font color="Green" title="E33">A28</font></span></a>, <a class="txt" href="#E5:15_2"><span class="lab"><font color="Green" title="E34">A31</font></span></a>, <a class="txt" href="#E8:15_2"><span class="lab"><font color="Green" title="E37">A34</font></span></a>, <a class="txt" href="#E6:15_2"><span class="lab"><font color="Green" title="E35">A32</font></span></a>, <a class="ref" href="finseq_3.html#T120" title="FINSEQ_3:th.120">FINSEQ_3:120</a>, <a class="ref" href="matrix_1.html#T25" title="MATRIX_1:th.25">MATRIX_1:25</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:15_2"/>
<span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">K</font>,<font color="Maroon" title="c14">U</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c13">K</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">U</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:15"><span class="lab"><font color="Green" title="E30">A16</font></span></a>, <a class="txt" href="#E7:15_2"><span class="lab"><font color="Green" title="E36">A33</font></span></a>, <a class="ref" href="matrix_1.html#D8" title="MATRIX_1:def.8">MATRIX_1:def 8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E27:15"/>
( ( <font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Maroon" title="c10">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">j</font> implies <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c10">m</font> ) &amp; ( <font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Maroon" title="c10">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">j</font> implies <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">k</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> ) &amp; ( <font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Maroon" title="c10">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">j</font> implies <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c10">m</font> ) &amp; ( <font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c4">i</font> &amp; <font color="Maroon" title="c10">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c5">j</font> implies <span class="p1">(<span class="default"><a href="laplace.html#K2" title="LAPLACE:func.2">Delete</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c5">j</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">k</font>,<font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">M</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:15"><span class="lab"><font color="Green" title="E31">A17</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>
