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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">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>, <font color="Olive" title="b2">y</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">n</font></span>)</span><br/>  for <font color="Olive" title="b3">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> (<font color="Olive" title="b3">i</font>,<font color="Maroon" title="c1">n</font>)</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">x</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> (<font color="Olive" title="b3">i</font>,<font color="Maroon" title="c1">n</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> <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="Olive" title="b3">i</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">y</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</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">n</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="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> (<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> (<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">x</font></span>)</span> <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="Olive" title="b1">i</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">y</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">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"> <span class="p1">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> (<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> (<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">x</font></span>)</span> <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="c4">i</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">y</font></span>)</span></span><br/>





<span class="kw">reconsider </span><font color="Maroon" title="c5">x1</font> = <font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c6">y1</font> = <font color="Maroon" title="c3">y</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">n</font> <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="c7">rx</font> = <font color="Maroon" title="c5">x1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">i</font>, <font color="Maroon" title="c8">ry</font> = <font color="Maroon" title="c6">y1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">i</font> as    <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>  ;<br/>
<a NAME="E3:11"/>
( <span class="p1">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> (<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">x</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"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c2">x</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> &amp; <span class="p1">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> (<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">y</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"><a href="pdiff_1.html#K1" title="PDIFF_1:func.1">proj</a> (<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">y</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="ref" href="pdiff_1.html#D4" title="PDIFF_1:def.4">PDIFF_1:def 4</a></span>;<br/>
<a NAME="E4:11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E8">A1</font></span>: 
( <span class="p1">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> (<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">x</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="c7">rx</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> &amp; <span class="p1">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> (<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">y</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="c8">ry</font></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="ref" href="pdiff_1.html#D1" title="PDIFF_1:def.1">PDIFF_1:def 1</a></span>;<br/>
<a NAME="E5:11"/><span class="lab"><font color="Green" title="E9">A2</font></span>: 
( <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c7">rx</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> is    <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 &amp; <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c8">ry</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> is    <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="finseq_2.html#T98" title="FINSEQ_2:th.98">FINSEQ_2:98</a></span>;<br/>
<span class="p1">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> (<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c3">y</font></span>)</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="c4">i</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c3">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="ref" href="pdiff_1.html#D4" title="PDIFF_1:def.4">PDIFF_1:def 4</a></span>
<br/>.= 
<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="c4">i</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c5">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">y1</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="ref" href="real_ns1.html#T5" title="REAL_NS1:th.5">REAL_NS1:5</a></span>
<br/>.= 
<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"><font color="Maroon" title="c5">x1</font> <a href="euclid.html#K8" title="EUCLID:func.8">-</a> <font color="Maroon" title="c6">y1</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">i</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="ref" href="pdiff_1.html#D1" title="PDIFF_1:def.1">PDIFF_1:def 1</a></span>
<br/>.= 
<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"><font color="Maroon" title="c5">x1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">i</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">y1</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">i</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="ref" href="rvsum_1.html#T27" title="RVSUM_1:th.27">RVSUM_1:27</a></span>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c7">rx</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="rvsum_1.html#K8" title="RVSUM_1:func.8">-</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c8">ry</font></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="ref" href="rvsum_1.html#T29" title="RVSUM_1:th.29">RVSUM_1:29</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E7:11"/>
<span class="p1">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> (<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Maroon" title="c3">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pdiff_1.html#K4" title="PDIFF_1:func.4">Proj</a> (<font color="Maroon" title="c4">i</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">x</font></span>)</span> <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="c4">i</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:11"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="txt" href="#E5:11"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="ref" href="real_ns1.html#T5" title="REAL_NS1:th.5">REAL_NS1:5</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>
