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<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <a href="ndiff_5.html#V1" title="NDIFF_5:attr.1">non-trivial</a>  <a href="prvect_2.html#NM3" title="PRVECT_2:NM.3">RealNormSpace-Sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">F</font> being   non  <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>  <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a><br/>  for <font color="Olive" title="b2">f1</font>, <font color="Olive" title="b3">f2</font> being   <a href="struct_0.html#NM12" title="STRUCT_0:NM.12">PartFunc</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span>,<font color="Olive" title="b1">F</font><br/>  for <font color="Olive" title="b4">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span><br/>  for <font color="Olive" title="b5">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font> holds <br/>( <span class="p1">(<span class="default"><font color="Olive" title="b2">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Olive" title="b3">f2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p2">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Olive" title="b5">i</font>)</span>)</span>,<font color="Olive" title="b4">x</font>)</span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">f1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Olive" title="b5">i</font>)</span>)</span>,<font color="Olive" title="b4">x</font>)</span>)</span></span>)</span> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Olive" title="b5">i</font>)</span>)</span>,<font color="Olive" title="b4">x</font>)</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b2">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Olive" title="b3">f2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p2">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Olive" title="b5">i</font>)</span>)</span>,<font color="Olive" title="b4">x</font>)</span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">f1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Olive" title="b5">i</font>)</span>)</span>,<font color="Olive" title="b4">x</font>)</span>)</span></span>)</span> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Olive" title="b5">i</font>)</span>)</span>,<font color="Olive" title="b4">x</font>)</span>)</span></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">F</font> be   non  <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>  <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f1</font>, <font color="Olive" title="b2">f2</font> being   <a href="struct_0.html#NM12" title="STRUCT_0:NM.12">PartFunc</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span>,<font color="Maroon" title="c2">F</font><br/>  for <font color="Olive" title="b3">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span><br/>  for <font color="Olive" title="b4">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b4">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font> holds <br/>( <span class="p1">(<span class="default"><font color="Olive" title="b1">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Olive" title="b2">f2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p2">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Olive" title="b4">i</font>)</span>)</span>,<font color="Olive" title="b3">x</font>)</span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Olive" title="b4">i</font>)</span>)</span>,<font color="Olive" title="b3">x</font>)</span>)</span></span>)</span> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Olive" title="b4">i</font>)</span>)</span>,<font color="Olive" title="b3">x</font>)</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Olive" title="b2">f2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p2">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Olive" title="b4">i</font>)</span>)</span>,<font color="Olive" title="b3">x</font>)</span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">f1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Olive" title="b4">i</font>)</span>)</span>,<font color="Olive" title="b3">x</font>)</span>)</span></span>)</span> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Olive" title="b4">i</font>)</span>)</span>,<font color="Olive" title="b3">x</font>)</span>)</span></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">f1</font>, <font color="Maroon" title="c4">f2</font> be   <a href="struct_0.html#NM12" title="STRUCT_0:NM.12">PartFunc</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span>,<font color="Maroon" title="c2">F</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span><br/>  for <font color="Olive" title="b2">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font> holds <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p2">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Olive" title="b2">i</font>)</span>)</span>,<font color="Olive" title="b1">x</font>)</span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Olive" title="b2">i</font>)</span>)</span>,<font color="Olive" title="b1">x</font>)</span>)</span></span>)</span> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Olive" title="b2">i</font>)</span>)</span>,<font color="Olive" title="b1">x</font>)</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p2">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Olive" title="b2">i</font>)</span>)</span>,<font color="Olive" title="b1">x</font>)</span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Olive" title="b2">i</font>)</span>)</span>,<font color="Olive" title="b1">x</font>)</span>)</span></span>)</span> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Olive" title="b2">i</font>)</span>)</span>,<font color="Olive" title="b1">x</font>)</span>)</span></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="prvect_2.html#K14" title="PRVECT_2:func.14">product</a> <font color="Maroon" title="c1">G</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">i</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font> holds <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p2">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Olive" title="b1">i</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Olive" title="b1">i</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Olive" title="b1">i</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p2">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Olive" title="b1">i</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Olive" title="b1">i</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Olive" title="b1">i</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">i0</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">i0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font> implies ( <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p2">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p2">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> ) )</span><br/>











<span class="kw">assume </span><a NAME="E1:45"/>
<font color="Maroon" title="c6">i0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p2">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p2">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c7">i</font> =  <a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>);<br/>
<a NAME="E2:45"/><span class="lab"><font color="Green" title="E38">A1</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p2">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E3:45"/><span class="lab"><font color="Green" title="E39">A2</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vfunct_1.html#D1" title="VFUNCT_1:def.1">VFUNCT_1:def 1</a></span>;<br/>
<a NAME="E4:45"/>
 for <font color="Olive" title="b1">s</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span></span>)</span> holds <br/> ( <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> iff <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p4">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p4">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">s</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> iff <font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p4">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p4">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span></span>)</span> )</span><br/>

<a NAME="E1:45_1"/>
( <font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> iff <span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p2">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f2</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:45"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="txt" href="#E2:45"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="ref" href="funct_1.html#T11" title="FUNCT_1:th.11">FUNCT_1:11</a></span>;<br/>
<a NAME="E2:45_1"/><span class="kw">then </span>
( <font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> iff ( <span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p2">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f1</font> &amp; <span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p2">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f2</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E3:45_1"/><span class="kw">then </span>
( <font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> iff ( <font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> &amp; <font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="ref" href="funct_1.html#T11" title="FUNCT_1:th.11">FUNCT_1:11</a></span>;<br/>
<a NAME="E4:45_1"/><span class="kw">then </span>
( <font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> iff <font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p4">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p4">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:45_1"/>
( <font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> iff <font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p4">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p4">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vfunct_1.html#D1" title="VFUNCT_1:def.1">VFUNCT_1:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:45"/><span class="kw">then </span>
 for <font color="Olive" title="b1">s</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> iff <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p4">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p4">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span></span>)</span> )
 
;<br/>
<a NAME="E6:45"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A3</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p4">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p4">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T1" title="TARSKI:th.1">TARSKI:1</a></span>;<br/>
<a NAME="E7:45"/><span class="lab"><font color="Green" title="E41">A4</font></span>: 
 for <font color="Olive" title="b1">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span></span>)</span>  st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p4">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p4">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p4">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p4">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:45_2"/><span class="lab"><font color="Green" title="E42">A5</font></span>: 
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p4">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p4">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font></span><br/>

<a NAME="E2:45_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A6</font></span>: 
<span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p2">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c4">f2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T11" title="FUNCT_1:th.11">FUNCT_1:11</a></span>;<br/>
<a NAME="E3:45_2"/>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p4">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p4">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:45"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="txt" href="#E1:45_2"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="ref" href="vfunct_1.html#D1" title="VFUNCT_1:def.1">VFUNCT_1:def 1</a></span>;<br/>
<a NAME="E4:45_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A9</font></span>: 
( <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> &amp; <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E5:45_2"/><span class="lab"><font color="Green" title="E45">A7</font></span>: 
<span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p2">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:45"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="txt" href="#E1:45_2"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="ref" href="funct_1.html#T11" title="FUNCT_1:th.11">FUNCT_1:11</a></span>;<br/>
<a NAME="E6:45_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p2">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E46">A13</font></span>: <font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">z</font></span>)</span> = 
<font color="Maroon" title="c3">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:45_2"><span class="lab"><font color="Green" title="E44">A9</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:45_2"><span class="lab"><font color="Green" title="E44">A9</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
;<br/>
<a NAME="E8:45_2"/>
<span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p2">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:45_2"><span class="lab"><font color="Green" title="E45">A7</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E47">A12</font></span>: <font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">z</font></span>)</span> = 
<font color="Maroon" title="c4">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:45_2"><span class="lab"><font color="Green" title="E44">A9</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:45_2"><span class="lab"><font color="Green" title="E44">A9</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_2"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45_2"><span class="lab"><font color="Green" title="E43">A6</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p4">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">z</font></span>)</span></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p4">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">z</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45_2"><span class="lab"><font color="Green" title="E43">A6</font></span></a>, <a class="ref" href="vfunct_1.html#D1" title="VFUNCT_1:def.1">VFUNCT_1:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p4">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p4">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:45"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="txt" href="#E1:45_2"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="txt" href="#E7:45_2"><span class="lab"><font color="Green" title="E46">A13</font></span></a>, <a class="txt" href="#E9:45_2"><span class="lab"><font color="Green" title="E47">A12</font></span></a>, <a class="ref" href="vfunct_1.html#D1" title="VFUNCT_1:def.1">VFUNCT_1:def 1</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E11:45_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p4">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p4">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:45"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="txt" href="#E1:45_2"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E8:45"/><span class="lab"><font color="Green" title="E42">A14</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="vfunct_1.html#D2" title="VFUNCT_1:def.2">VFUNCT_1:def 2</a></span>;<br/>
<a NAME="E9:45"/>
 for <font color="Olive" title="b1">s</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span></span>)</span> holds <br/> ( <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> iff <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p4">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p4">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">s</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> iff <font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p4">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p4">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span></span>)</span> )</span><br/>

<a NAME="E1:45_3"/>
( <font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> iff <span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p2">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f2</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:45"><span class="lab"><font color="Green" title="E42">A14</font></span></a>, <a class="txt" href="#E2:45"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="ref" href="funct_1.html#T11" title="FUNCT_1:th.11">FUNCT_1:11</a></span>;<br/>
<a NAME="E2:45_3"/><span class="kw">then </span>
( <font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> iff ( <span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p2">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f1</font> &amp; <span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p2">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f2</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E3:45_3"/><span class="kw">then </span>
( <font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> iff ( <font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> &amp; <font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="ref" href="funct_1.html#T11" title="FUNCT_1:th.11">FUNCT_1:11</a></span>;<br/>
<a NAME="E4:45_3"/><span class="kw">then </span>
( <font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> iff <font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p4">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p4">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:45_3"/>
( <font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> iff <font color="Maroon" title="c8">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p4">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p4">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="vfunct_1.html#D2" title="VFUNCT_1:def.2">VFUNCT_1:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E10:45"/><span class="kw">then </span>
 for <font color="Olive" title="b1">s</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> iff <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p4">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p4">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span></span>)</span> )
 
;<br/>
<a NAME="E11:45"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A15</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p4">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p4">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T1" title="TARSKI:th.1">TARSKI:1</a></span>;<br/>
<a NAME="E12:45"/>
 for <font color="Olive" title="b1">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span></span>)</span>  st <font color="Olive" title="b1">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p4">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p4">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="prvect_2.html#K11" title="PRVECT_2:func.11">.</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p4">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p4">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:45_4"/><span class="lab"><font color="Green" title="E44">A16</font></span>: 
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p4">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p4">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font></span><br/>

<a NAME="E2:45_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A17</font></span>: 
<span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p2">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T11" title="FUNCT_1:th.11">FUNCT_1:11</a></span>;<br/>
<a NAME="E3:45_4"/>
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p4">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p4">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:45"><span class="lab"><font color="Green" title="E43">A15</font></span></a>, <a class="txt" href="#E1:45_4"><span class="lab"><font color="Green" title="E44">A16</font></span></a>, <a class="ref" href="vfunct_1.html#D2" title="VFUNCT_1:def.2">VFUNCT_1:def 2</a></span>;<br/>
<a NAME="E4:45_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A20</font></span>: 
( <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> &amp; <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E5:45_4"/><span class="lab"><font color="Green" title="E47">A18</font></span>: 
<span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p2">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:45"><span class="lab"><font color="Green" title="E42">A14</font></span></a>, <a class="txt" href="#E1:45_4"><span class="lab"><font color="Green" title="E44">A16</font></span></a>, <a class="ref" href="funct_1.html#T11" title="FUNCT_1:th.11">FUNCT_1:11</a></span>;<br/>
<a NAME="E6:45_4"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p2">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c3">f1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E48">A24</font></span>: <font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">z</font></span>)</span> = 
<font color="Maroon" title="c3">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:45_4"><span class="lab"><font color="Green" title="E46">A20</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:45_4"><span class="lab"><font color="Green" title="E46">A20</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
;<br/>
<a NAME="E8:45_4"/>
<span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p2">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c4">f2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:45_4"><span class="lab"><font color="Green" title="E47">A18</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E49">A23</font></span>: <font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">z</font></span>)</span> = 
<font color="Maroon" title="c4">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:45_4"><span class="lab"><font color="Green" title="E46">A20</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:45_4"><span class="lab"><font color="Green" title="E46">A20</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45_4"><span class="lab"><font color="Green" title="E44">A16</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45_4"><span class="lab"><font color="Green" title="E45">A17</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p4">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">z</font></span>)</span></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p4">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">z</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45_4"><span class="lab"><font color="Green" title="E45">A17</font></span></a>, <a class="ref" href="vfunct_1.html#D2" title="VFUNCT_1:def.2">VFUNCT_1:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p4">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p4">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:45"><span class="lab"><font color="Green" title="E43">A15</font></span></a>, <a class="txt" href="#E1:45_4"><span class="lab"><font color="Green" title="E44">A16</font></span></a>, <a class="txt" href="#E7:45_4"><span class="lab"><font color="Green" title="E48">A24</font></span></a>, <a class="txt" href="#E9:45_4"><span class="lab"><font color="Green" title="E49">A23</font></span></a>, <a class="ref" href="vfunct_1.html#D2" title="VFUNCT_1:def.2">VFUNCT_1:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p4">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p4">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:45"><span class="lab"><font color="Green" title="E43">A15</font></span></a>, <a class="txt" href="#E1:45_4"><span class="lab"><font color="Green" title="E44">A16</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E13:45"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p2">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> <a href="vfunct_1.html#K6" title="VFUNCT_1:func.6">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <font color="Maroon" title="c4">f2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p2">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">f1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> <a href="vfunct_1.html#K2" title="VFUNCT_1:func.2">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="ndiff_5.html#K4" title="NDIFF_5:func.4">reproj</a> (<span class="p3">(<span class="default"><a href="ndiff_5.html#K5" title="NDIFF_5:func.5">modetrans</a> (<font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c6">i0</font>)</span>)</span>,<font color="Maroon" title="c5">x</font>)</span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:45"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="txt" href="#E11:45"><span class="lab"><font color="Green" title="E43">A15</font></span></a>, <a class="txt" href="#E7:45"><span class="lab"><font color="Green" title="E41">A4</font></span></a>, <a class="ref" href="partfun1.html#T5" title="PARTFUN1:th.5">PARTFUN1:5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
