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<span class="kw">let </span><font color="Maroon" title="c1">m</font>, <font color="Maroon" title="c2">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="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="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span><br/>  for <font color="Olive" title="b2">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 1</span>)</span> holds <br/> ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c2">i</font>,<font color="Olive" title="b1">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">p</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c2">i</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">p</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span></span>)</span> &amp; <font color="Olive" title="b1">x</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c2">i</font>,<font color="Olive" title="b1">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c2">i</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Olive" title="b2">p</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">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="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</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">p</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 1</span>)</span> holds <br/> ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c2">i</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c2">i</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Olive" title="b1">p</font></span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> 1</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"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c2">i</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c2">i</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">p</font></span>)</span> )</span><br/>





<span class="kw">consider </span><font color="Maroon" title="c5">p1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> , <font color="Maroon" title="c6">y</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">m</font><span class="kw"> such that </span><br/><a NAME="E2:16"/><span class="lab"><font color="Green" title="E13">A1</font></span>: 
( <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c5">p1</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> &amp; <span class="p1">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c6">y</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pdiff_1.html#D6" title="PDIFF_1:def.6">PDIFF_1:def 6</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">pm</font> = <font color="Maroon" title="c4">p</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 1 <span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D4" title="REAL_NS1:def.4">REAL_NS1:def 4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">rm</font> = <span class="p1">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c6">y</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c5">p1</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">m</font> ;<br/>
<a NAME="E5:16"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">rm</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="ref" href="real_ns1.html#T5" title="REAL_NS1:th.5">REAL_NS1:5</a></span>;<br/>
<a NAME="E6:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E14">A2</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (<font color="Maroon" title="c2">i</font>,<span class="p2">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">p1</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pdiff_7.html#T6" target="_self" title="PDIFF_7:th.6">Th6</a></span>;<br/>
<a NAME="E7:16"/><span class="lab"><font color="Green" title="E15">A3</font></span>: 
 <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D4" title="REAL_NS1:def.4">REAL_NS1:def 4</a></span>;<br/>
<a NAME="E8:16"/><span class="lab"><font color="Green" title="E16">A4</font></span>: 
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">y</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="ref" href="pdiff_1.html#D4" title="PDIFF_1:def.4">PDIFF_1:def 4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">Pr</font> = <span class="p1">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> 1 <span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#D4" title="REAL_NS1:def.4">REAL_NS1:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">r1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> , <font color="Maroon" title="c11">z</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">m</font><span class="kw"> such that </span><br/><a NAME="E11:16"/><span class="lab"><font color="Green" title="E17">A5</font></span>: 
( <font color="Maroon" title="c4">p</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c10">r1</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> &amp; <font color="Maroon" title="c11">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c2">i</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c11">z</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">r1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pdiff_1.html#D6" title="PDIFF_1:def.6">PDIFF_1:def 6</a></span>;<br/>
<a NAME="E12:16"/>
<font color="Maroon" title="c4">p</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">pm</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c9">Pr</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#T5" title="REAL_NS1:th.5">REAL_NS1:5</a></span>;<br/>
<a NAME="E13:16"/><span class="kw">then </span>
<font color="Maroon" title="c4">p</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">p1</font> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">y</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="txt" href="#E8:16"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="ref" href="rvsum_1.html#T29" title="RVSUM_1:th.29">RVSUM_1:29</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:16"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c2">i</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">p</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:16"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="txt" href="#E7:16"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="txt" href="#E11:16"><span class="lab"><font color="Green" title="E17">A5</font></span></a>, <a class="ref" href="finseq_1.html#T76" title="FINSEQ_1:th.76">FINSEQ_1:76</a></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="c3">x</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c2">i</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">p</font></span>)</span></span><br/>

<a NAME="E15:16"/>
<font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c8">rm</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="ref" href="real_ns1.html#T5" title="REAL_NS1:th.5">REAL_NS1:5</a></span>;<br/>
<a NAME="E16:16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A6</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (<font color="Maroon" title="c2">i</font>,<span class="p2">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">m</font></span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c5">p1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pdiff_7.html#T6" target="_self" title="PDIFF_7:th.6">Th6</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c12">r2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> , <font color="Maroon" title="c13">z</font> being    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">m</font><span class="kw"> such that </span><br/><a NAME="E18:16"/><span class="lab"><font color="Green" title="E19">A7</font></span>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c12">r2</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> &amp; <font color="Maroon" title="c13">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c2">i</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K6" title="PDIFF_1:func.6">reproj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c13">z</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">r2</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pdiff_1.html#D6" title="PDIFF_1:def.6">PDIFF_1:def 6</a></span>;<br/>
<a NAME="E19:16"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">Pr</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c7">pm</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="real_ns1.html#T5" title="REAL_NS1:th.5">REAL_NS1:5</a></span>;<br/>
<a NAME="E20:16"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c6">y</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <font color="Maroon" title="c5">p1</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:16"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="txt" href="#E8:16"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="ref" href="rvsum_1.html#T29" title="RVSUM_1:th.29">RVSUM_1:29</a></span>;<br/>
<span class="kw">hence </span><a NAME="E21:16"/>
<font color="Maroon" title="c3">x</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">x</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="pdiff_1.html#K7" title="PDIFF_1:func.7">reproj</a> (<font color="Maroon" title="c2">i</font>,<span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="real_ns1.html#K4" title="REAL_NS1:func.4">REAL-NS</a> <font color="Maroon" title="c1">m</font></span>)</span></span>)</span>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> (<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">m</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c4">p</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:16"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="txt" href="#E18:16"><span class="lab"><font color="Green" title="E19">A7</font></span></a>, <a class="txt" href="#E16:16"><span class="lab"><font color="Green" title="E18">A6</font></span></a>, <a class="ref" href="finseq_1.html#T76" title="FINSEQ_1:th.76">FINSEQ_1:76</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/>


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