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<b>let </b><font color="Maroon" title="c1">seq</font> be   <a href="mesfunc5.html#NM1" title="MESFUNC5:NM.1">ExtREAL_sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">rseq</font> being  <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>  st <font color="Maroon" title="c1">seq</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b1">rseq</font> holds <br/>( ( <font color="Olive" title="b1">rseq</font> is <a href="seqm_3.html#V4" title="SEQM_3:attr.4">non-increasing</a> implies <font color="Maroon" title="c1">seq</font> is <a href="rinfsup2.html#V7" title="RINFSUP2:attr.7">non-increasing</a> ) &amp; ( <font color="Maroon" title="c1">seq</font> is <a href="rinfsup2.html#V7" title="RINFSUP2:attr.7">non-increasing</a> implies <font color="Olive" title="b1">rseq</font> is <a href="seqm_3.html#V4" title="SEQM_3:attr.4">non-increasing</a> ) &amp; ( <font color="Olive" title="b1">rseq</font> is <a href="seqm_3.html#V3" title="SEQM_3:attr.3">non-decreasing</a> implies <font color="Maroon" title="c1">seq</font> is <a href="rinfsup2.html#V6" title="RINFSUP2:attr.6">non-decreasing</a> ) &amp; ( <font color="Maroon" title="c1">seq</font> is <a href="rinfsup2.html#V6" title="RINFSUP2:attr.6">non-decreasing</a> implies <font color="Olive" title="b1">rseq</font> is <a href="seqm_3.html#V3" title="SEQM_3:attr.3">non-decreasing</a> ) )</span><br/><b>let </b><font color="Maroon" title="c2">rseq</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">seq</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c2">rseq</font> implies ( ( <font color="Maroon" title="c2">rseq</font> is <a href="seqm_3.html#V4" title="SEQM_3:attr.4">non-increasing</a> implies <font color="Maroon" title="c1">seq</font> is <a href="rinfsup2.html#V7" title="RINFSUP2:attr.7">non-increasing</a> ) &amp; ( <font color="Maroon" title="c1">seq</font> is <a href="rinfsup2.html#V7" title="RINFSUP2:attr.7">non-increasing</a> implies <font color="Maroon" title="c2">rseq</font> is <a href="seqm_3.html#V4" title="SEQM_3:attr.4">non-increasing</a> ) &amp; ( <font color="Maroon" title="c2">rseq</font> is <a href="seqm_3.html#V3" title="SEQM_3:attr.3">non-decreasing</a> implies <font color="Maroon" title="c1">seq</font> is <a href="rinfsup2.html#V6" title="RINFSUP2:attr.6">non-decreasing</a> ) &amp; ( <font color="Maroon" title="c1">seq</font> is <a href="rinfsup2.html#V6" title="RINFSUP2:attr.6">non-decreasing</a> implies <font color="Maroon" title="c2">rseq</font> is <a href="seqm_3.html#V3" title="SEQM_3:attr.3">non-decreasing</a> ) ) )</span><br/>



<b>assume </b><a NAME="E1:24"/>
<font color="Maroon" title="c1">seq</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c2">rseq</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c2">rseq</font> is <a href="seqm_3.html#V4" title="SEQM_3:attr.4">non-increasing</a> implies <font color="Maroon" title="c1">seq</font> is <a href="rinfsup2.html#V7" title="RINFSUP2:attr.7">non-increasing</a> ) &amp; ( <font color="Maroon" title="c1">seq</font> is <a href="rinfsup2.html#V7" title="RINFSUP2:attr.7">non-increasing</a> implies <font color="Maroon" title="c2">rseq</font> is <a href="seqm_3.html#V4" title="SEQM_3:attr.4">non-increasing</a> ) &amp; ( <font color="Maroon" title="c2">rseq</font> is <a href="seqm_3.html#V3" title="SEQM_3:attr.3">non-decreasing</a> implies <font color="Maroon" title="c1">seq</font> is <a href="rinfsup2.html#V6" title="RINFSUP2:attr.6">non-decreasing</a> ) &amp; ( <font color="Maroon" title="c1">seq</font> is <a href="rinfsup2.html#V6" title="RINFSUP2:attr.6">non-decreasing</a> implies <font color="Maroon" title="c2">rseq</font> is <a href="seqm_3.html#V3" title="SEQM_3:attr.3">non-decreasing</a> ) )</span><br/>

<a NAME="E2:24"/><b>then </b><i><font color="Green" title="E25">A1</font></i>: 
( ( ( for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</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>   st <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> holds <br/><font color="Maroon" title="c1">seq</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">seq</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font> ) implies for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</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>   st <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> holds <br/><font color="Maroon" title="c2">rseq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">rseq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> ) &amp; ( ( for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</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>   st <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> holds <br/><font color="Maroon" title="c2">rseq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">rseq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> ) implies for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</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>   st <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> holds <br/><font color="Maroon" title="c1">seq</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">seq</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font> ) &amp; ( ( for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</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>   st <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> holds <br/><font color="Maroon" title="c1">seq</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">seq</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b2">m</font> ) implies for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</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>   st <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> holds <br/><font color="Maroon" title="c2">rseq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">rseq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">m</font> ) &amp; ( ( for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</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>   st <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> holds <br/><font color="Maroon" title="c2">rseq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">rseq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">m</font> ) implies for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</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>   st <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> holds <br/><font color="Maroon" title="c1">seq</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">seq</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Olive" title="b2">m</font> ) )
 
;<br/>
<a NAME="E3:24"/>
( ( <font color="Maroon" title="c2">rseq</font> is <a href="seqm_3.html#V3" title="SEQM_3:attr.3">non-decreasing</a> implies for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</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>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Maroon" title="c2">rseq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">rseq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">m</font> ) &amp; ( ( for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</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>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Maroon" title="c2">rseq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">rseq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">m</font> ) implies <font color="Maroon" title="c2">rseq</font> is <a href="seqm_3.html#V3" title="SEQM_3:attr.3">non-decreasing</a> ) &amp; ( <font color="Maroon" title="c2">rseq</font> is <a href="seqm_3.html#V4" title="SEQM_3:attr.4">non-increasing</a> implies for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</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>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Maroon" title="c2">rseq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">rseq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> ) &amp; ( ( for <font color="Olive" title="b1">n</font>, <font color="Olive" title="b2">m</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>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><font color="Maroon" title="c2">rseq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">rseq</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> ) implies <font color="Maroon" title="c2">rseq</font> is <a href="seqm_3.html#V4" title="SEQM_3:attr.4">non-increasing</a> ) )
 
<b>by </b><i><a class="ref" href="seqm_3.html#T12" title="SEQM_3:th.12">SEQM_3:12</a>, <a class="ref" href="seqm_3.html#T14" title="SEQM_3:th.14">SEQM_3:14</a></i>;<br/>
<b>hence </b><a NAME="E4:24"/>
( ( <font color="Maroon" title="c2">rseq</font> is <a href="seqm_3.html#V4" title="SEQM_3:attr.4">non-increasing</a> implies <font color="Maroon" title="c1">seq</font> is <a href="rinfsup2.html#V7" title="RINFSUP2:attr.7">non-increasing</a> ) &amp; ( <font color="Maroon" title="c1">seq</font> is <a href="rinfsup2.html#V7" title="RINFSUP2:attr.7">non-increasing</a> implies <font color="Maroon" title="c2">rseq</font> is <a href="seqm_3.html#V4" title="SEQM_3:attr.4">non-increasing</a> ) &amp; ( <font color="Maroon" title="c2">rseq</font> is <a href="seqm_3.html#V3" title="SEQM_3:attr.3">non-decreasing</a> implies <font color="Maroon" title="c1">seq</font> is <a href="rinfsup2.html#V6" title="RINFSUP2:attr.6">non-decreasing</a> ) &amp; ( <font color="Maroon" title="c1">seq</font> is <a href="rinfsup2.html#V6" title="RINFSUP2:attr.6">non-decreasing</a> implies <font color="Maroon" title="c2">rseq</font> is <a href="seqm_3.html#V3" title="SEQM_3:attr.3">non-decreasing</a> ) )
 <b>by </b><i><a class="txt" href="rinfsup2.html#E2:24"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="rinfsup2.html#T7" target="_self" title="RINFSUP2:th.7">Th7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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