let A be closed-interval Subset of REAL ; :: thesis: for D2 being Division of A st lower_bound A < D2 . 1 holds
<*(lower_bound A)*> ^ D2 is non empty increasing FinSequence of REAL
let D2 be Division of A; :: thesis: ( lower_bound A < D2 . 1 implies <*(lower_bound A)*> ^ D2 is non empty increasing FinSequence of REAL )
assume A1: lower_bound A < D2 . 1 ; :: thesis: <*(lower_bound A)*> ^ D2 is non empty increasing FinSequence of REAL
set MD2 = <*(lower_bound A)*> ^ D2;
for n, m being Element of NAT st n in dom (<*(lower_bound A)*> ^ D2) & m in dom (<*(lower_bound A)*> ^ D2) & n < m holds
(<*(lower_bound A)*> ^ D2) . n < (<*(lower_bound A)*> ^ D2) . m

proof

let n, m be Element of NAT ; :: thesis: ( n in dom (<*(lower_bound A)*> ^ D2) & m in dom (<*(lower_bound A)*> ^ D2) & n < m implies (<*(lower_bound A)*> ^ D2) . n < (<*(lower_bound A)*> ^ D2) . m )
assume A2: ( n in dom (<*(lower_bound A)*> ^ D2) & m in dom (<*(lower_bound A)*> ^ D2) & n < m ) ; :: thesis: (<*(lower_bound A)*> ^ D2) . n < (<*(lower_bound A)*> ^ D2) . m
A3: not m in dom <*(lower_bound A)*>

proof

assume m in dom <*(lower_bound A)*> ; :: thesis: contradiction
then m in Seg (len <*(lower_bound A)*>) by FINSEQ_1:def 3;
then m in {1} by FINSEQ_1:4, FINSEQ_1:56;
then A4: n < 1 by A2, TARSKI:def 1;
n in Seg (len (<*(lower_bound A)*> ^ D2)) by A2, FINSEQ_1:def 3;
hence contradiction by A4, FINSEQ_1:3; :: thesis: verum

end;

(<*(lower_bound A)*> ^ D2) . m in rng (<*(lower_bound A)*> ^ D2) by A2, FUNCT_1:def 5;
then (<*(lower_bound A)*> ^ D2) . m in (rng <*(lower_bound A)*>) \/ (rng D2) by FINSEQ_1:44;
then A5: ( (<*(lower_bound A)*> ^ D2) . m in rng <*(lower_bound A)*> or (<*(lower_bound A)*> ^ D2) . m in rng D2 ) by XBOOLE_0:def 3;
A6: not (<*(lower_bound A)*> ^ D2) . m in rng <*(lower_bound A)*>

proof

assume (<*(lower_bound A)*> ^ D2) . m in rng <*(lower_bound A)*> ; :: thesis: contradiction
then (<*(lower_bound A)*> ^ D2) . m in {(lower_bound A)} by FINSEQ_1:55;
then A7: (<*(lower_bound A)*> ^ D2) . m = lower_bound A by TARSKI:def 1;
consider n being Nat such that
A8: ( n in dom D2 & m = (len <*(lower_bound A)*>) + n ) by A2, A3, FINSEQ_1:38;
n in Seg (len D2) by A8, FINSEQ_1:def 3;
then A9: ( 1 <= n & n <= len D2 ) by FINSEQ_1:3;
rng D2 <> {} ;
then A10: 1 in dom D2 by FINSEQ_3:34;
D2 . n = (<*(lower_bound A)*> ^ D2) . m by A8, FINSEQ_1:def 7;
hence contradiction by A1, A7, A8, A9, A10, GOBOARD2:18; :: thesis: verum

end;

now

per cases ( n in dom <*(lower_bound A)*> or ex i being Nat st
( i in dom D2 & n = (len <*(lower_bound A)*>) + i ) ) by A2, FINSEQ_1:38;

suppose A11: n in dom <*(lower_bound A)*> ; :: thesis: (<*(lower_bound A)*> ^ D2) . n < (<*(lower_bound A)*> ^ D2) . m

then n in Seg (len <*(lower_bound A)*>) by FINSEQ_1:def 3;
then n in {1} by FINSEQ_1:4, FINSEQ_1:56;
then A12: n = 1 by TARSKI:def 1;
consider k being Element of NAT such that
A13: ( k in dom D2 & (<*(lower_bound A)*> ^ D2) . m = D2 . k ) by A5, A6, PARTFUN1:26;
k in Seg (len D2) by A13, FINSEQ_1:def 3;
then A14: ( 1 <= k & k <= len D2 ) by FINSEQ_1:3;
rng D2 <> {} ;
then 1 in dom D2 by FINSEQ_3:34;
then A15: D2 . 1 <= (<*(lower_bound A)*> ^ D2) . m by A13, A14, GOBOARD2:18;
(<*(lower_bound A)*> ^ D2) . n = <*(lower_bound A)*> . n by A11, FINSEQ_1:def 7
.= lower_bound A by A12, FINSEQ_1:def 8 ;
hence (<*(lower_bound A)*> ^ D2) . n < (<*(lower_bound A)*> ^ D2) . m by A1, A15, XXREAL_0:2; :: thesis: verum

end;

suppose ex i being Nat st
( i in dom D2 & n = (len <*(lower_bound A)*>) + i ) ; :: thesis: (<*(lower_bound A)*> ^ D2) . n < (<*(lower_bound A)*> ^ D2) . m

then consider i being Element of NAT such that
A16: ( i in dom D2 & n = (len <*(lower_bound A)*>) + i ) ;
consider j being Nat such that
A17: ( j in dom D2 & m = (len <*(lower_bound A)*>) + j ) by A2, A3, FINSEQ_1:38;
A18: i < j by A2, A16, A17, XREAL_1:8;
A19: D2 . i = (<*(lower_bound A)*> ^ D2) . n by A16, FINSEQ_1:def 7;
D2 . j = (<*(lower_bound A)*> ^ D2) . m by A17, FINSEQ_1:def 7;
hence (<*(lower_bound A)*> ^ D2) . n < (<*(lower_bound A)*> ^ D2) . m by A16, A17, A18, A19, SEQM_3:def 1; :: thesis: verum

end;

end;

end;

hence (<*(lower_bound A)*> ^ D2) . n < (<*(lower_bound A)*> ^ D2) . m ; :: thesis: verum

end;

hence <*(lower_bound A)*> ^ D2 is non empty increasing FinSequence of REAL by SEQM_3:def 1; :: thesis: verum