let A be non empty 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 )

reconsider lb = lower_bound A as Element of REAL by XREAL_0:def 1;

set MD2 = <*lb*> ^ D2;

assume A1: lower_bound A < D2 . 1 ; :: thesis: <*(lower_bound A)*> ^ D2 is non empty increasing FinSequence of REAL

for n, m being Nat st n in dom (<*lb*> ^ D2) & m in dom (<*lb*> ^ D2) & n < m holds

(<*lb*> ^ D2) . n < (<*lb*> ^ D2) . m

<*(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 )

reconsider lb = lower_bound A as Element of REAL by XREAL_0:def 1;

set MD2 = <*lb*> ^ D2;

assume A1: lower_bound A < D2 . 1 ; :: thesis: <*(lower_bound A)*> ^ D2 is non empty increasing FinSequence of REAL

for n, m being Nat st n in dom (<*lb*> ^ D2) & m in dom (<*lb*> ^ D2) & n < m holds

(<*lb*> ^ D2) . n < (<*lb*> ^ D2) . m

proof

hence
<*(lower_bound A)*> ^ D2 is non empty increasing FinSequence of REAL
by SEQM_3:def 1; :: thesis: verum
let n, m be Nat; :: thesis: ( n in dom (<*lb*> ^ D2) & m in dom (<*lb*> ^ D2) & n < m implies (<*lb*> ^ D2) . n < (<*lb*> ^ D2) . m )

assume that

A2: n in dom (<*lb*> ^ D2) and

A3: m in dom (<*lb*> ^ D2) and

A4: n < m ; :: thesis: (<*lb*> ^ D2) . n < (<*lb*> ^ D2) . m

A5: not m in dom <*(lower_bound A)*>

then (<*lb*> ^ D2) . m in (rng <*(lower_bound A)*>) \/ (rng D2) by FINSEQ_1:31;

then A13: ( (<*lb*> ^ D2) . m in rng <*(lower_bound A)*> or (<*lb*> ^ D2) . m in rng D2 ) by XBOOLE_0:def 3;

end;assume that

A2: n in dom (<*lb*> ^ D2) and

A3: m in dom (<*lb*> ^ D2) and

A4: n < m ; :: thesis: (<*lb*> ^ D2) . n < (<*lb*> ^ D2) . m

A5: not m in dom <*(lower_bound A)*>

proof

A7:
not (<*lb*> ^ D2) . m in rng <*(lower_bound A)*>
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_1:2, FINSEQ_1:39;

then A6: n < 1 by A4, TARSKI:def 1;

n in Seg (len (<*lb*> ^ D2)) by A2, FINSEQ_1:def 3;

hence contradiction by A6, FINSEQ_1:1; :: thesis: verum

end;then m in Seg (len <*(lower_bound A)*>) by FINSEQ_1:def 3;

then m in {1} by FINSEQ_1:2, FINSEQ_1:39;

then A6: n < 1 by A4, TARSKI:def 1;

n in Seg (len (<*lb*> ^ D2)) by A2, FINSEQ_1:def 3;

hence contradiction by A6, FINSEQ_1:1; :: thesis: verum

proof

(<*lb*> ^ D2) . m in rng (<*lb*> ^ D2)
by A3, FUNCT_1:def 3;
assume
(<*lb*> ^ D2) . m in rng <*(lower_bound A)*>
; :: thesis: contradiction

then (<*lb*> ^ D2) . m in {(lower_bound A)} by FINSEQ_1:38;

then A8: (<*lb*> ^ D2) . m = lower_bound A by TARSKI:def 1;

rng D2 <> {} ;

then A9: 1 in dom D2 by FINSEQ_3:32;

consider n being Nat such that

A10: n in dom D2 and

A11: m = (len <*(lower_bound A)*>) + n by A3, A5, FINSEQ_1:25;

n in Seg (len D2) by A10, FINSEQ_1:def 3;

then A12: 1 <= n by FINSEQ_1:1;

D2 . n = (<*lb*> ^ D2) . m by A10, A11, FINSEQ_1:def 7;

hence contradiction by A1, A8, A10, A12, A9, SEQ_4:137; :: thesis: verum

end;then (<*lb*> ^ D2) . m in {(lower_bound A)} by FINSEQ_1:38;

then A8: (<*lb*> ^ D2) . m = lower_bound A by TARSKI:def 1;

rng D2 <> {} ;

then A9: 1 in dom D2 by FINSEQ_3:32;

consider n being Nat such that

A10: n in dom D2 and

A11: m = (len <*(lower_bound A)*>) + n by A3, A5, FINSEQ_1:25;

n in Seg (len D2) by A10, FINSEQ_1:def 3;

then A12: 1 <= n by FINSEQ_1:1;

D2 . n = (<*lb*> ^ D2) . m by A10, A11, FINSEQ_1:def 7;

hence contradiction by A1, A8, A10, A12, A9, SEQ_4:137; :: thesis: verum

then (<*lb*> ^ D2) . m in (rng <*(lower_bound A)*>) \/ (rng D2) by FINSEQ_1:31;

then A13: ( (<*lb*> ^ D2) . m in rng <*(lower_bound A)*> or (<*lb*> ^ D2) . m in rng D2 ) by XBOOLE_0:def 3;

now :: thesis: (<*lb*> ^ D2) . n < (<*lb*> ^ D2) . mend;

hence
(<*lb*> ^ D2) . n < (<*lb*> ^ D2) . m
; :: thesis: verumper 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_1:25;

end;

( i in dom D2 & n = (len <*(lower_bound A)*>) + i ) ) by A2, FINSEQ_1:25;

suppose A14:
n in dom <*(lower_bound A)*>
; :: thesis: (<*lb*> ^ D2) . n < (<*lb*> ^ D2) . m

then
n in Seg (len <*(lower_bound A)*>)
by FINSEQ_1:def 3;

then n in {1} by FINSEQ_1:2, FINSEQ_1:39;

then A15: n = 1 by TARSKI:def 1;

A16: (<*lb*> ^ D2) . n = <*(lower_bound A)*> . n by A14, FINSEQ_1:def 7

.= lower_bound A by A15, FINSEQ_1:def 8 ;

rng D2 <> {} ;

then A17: 1 in dom D2 by FINSEQ_3:32;

consider k being Element of NAT such that

A18: k in dom D2 and

A19: (<*lb*> ^ D2) . m = D2 . k by A13, A7, PARTFUN1:3;

k in Seg (len D2) by A18, FINSEQ_1:def 3;

then 1 <= k by FINSEQ_1:1;

then D2 . 1 <= (<*lb*> ^ D2) . m by A18, A19, A17, SEQ_4:137;

hence (<*lb*> ^ D2) . n < (<*lb*> ^ D2) . m by A1, A16, XXREAL_0:2; :: thesis: verum

end;then n in {1} by FINSEQ_1:2, FINSEQ_1:39;

then A15: n = 1 by TARSKI:def 1;

A16: (<*lb*> ^ D2) . n = <*(lower_bound A)*> . n by A14, FINSEQ_1:def 7

.= lower_bound A by A15, FINSEQ_1:def 8 ;

rng D2 <> {} ;

then A17: 1 in dom D2 by FINSEQ_3:32;

consider k being Element of NAT such that

A18: k in dom D2 and

A19: (<*lb*> ^ D2) . m = D2 . k by A13, A7, PARTFUN1:3;

k in Seg (len D2) by A18, FINSEQ_1:def 3;

then 1 <= k by FINSEQ_1:1;

then D2 . 1 <= (<*lb*> ^ D2) . m by A18, A19, A17, SEQ_4:137;

hence (<*lb*> ^ D2) . n < (<*lb*> ^ D2) . m by A1, A16, XXREAL_0:2; :: thesis: verum

suppose
ex i being Nat st

( i in dom D2 & n = (len <*(lower_bound A)*>) + i ) ; :: thesis: (<*lb*> ^ D2) . n < (<*lb*> ^ D2) . m

( i in dom D2 & n = (len <*(lower_bound A)*>) + i ) ; :: thesis: (<*lb*> ^ D2) . n < (<*lb*> ^ D2) . m

then consider i being Element of NAT such that

A20: i in dom D2 and

A21: n = (len <*(lower_bound A)*>) + i ;

A22: D2 . i = (<*lb*> ^ D2) . n by A20, A21, FINSEQ_1:def 7;

consider j being Nat such that

A23: j in dom D2 and

A24: m = (len <*(lower_bound A)*>) + j by A3, A5, FINSEQ_1:25;

A25: D2 . j = (<*lb*> ^ D2) . m by A23, A24, FINSEQ_1:def 7;

i < j by A4, A21, A24, XREAL_1:6;

hence (<*lb*> ^ D2) . n < (<*lb*> ^ D2) . m by A20, A23, A22, A25, SEQM_3:def 1; :: thesis: verum

end;A20: i in dom D2 and

A21: n = (len <*(lower_bound A)*>) + i ;

A22: D2 . i = (<*lb*> ^ D2) . n by A20, A21, FINSEQ_1:def 7;

consider j being Nat such that

A23: j in dom D2 and

A24: m = (len <*(lower_bound A)*>) + j by A3, A5, FINSEQ_1:25;

A25: D2 . j = (<*lb*> ^ D2) . m by A23, A24, FINSEQ_1:def 7;

i < j by A4, A21, A24, XREAL_1:6;

hence (<*lb*> ^ D2) . n < (<*lb*> ^ D2) . m by A20, A23, A22, A25, SEQM_3:def 1; :: thesis: verum