let r be Real; :: thesis: for seq being Real_Sequence st seq is bounded_below holds
( r = lower_bound seq iff ( ( for n being Nat holds r <= seq . n ) & ( for s being Real st 0 < s holds
ex k being Nat st seq . k < r + s ) ) )
let seq be Real_Sequence; :: thesis: ( seq is bounded_below implies ( r = lower_bound seq iff ( ( for n being Nat holds r <= seq . n ) & ( for s being Real st 0 < s holds
ex k being Nat st seq . k < r + s ) ) ) )
set R = rng seq;
assume seq is bounded_below ; :: thesis: ( r = lower_bound seq iff ( ( for n being Nat holds r <= seq . n ) & ( for s being Real st 0 < s holds
ex k being Nat st seq . k < r + s ) ) )
then A1: rng seq is bounded_below by Th6;
A2: rng seq <> {} by RELAT_1:41;
A3: ( ( for n being Nat holds r <= seq . n ) & ( for s being Real st 0 < s holds
ex k being Nat st seq . k < r + s ) implies r = lower_bound seq )

proof

assume that
A4: for n being Nat holds r <= seq . n and
A5: for s being Real st 0 < s holds
ex k being Nat st seq . k < r + s ; :: thesis: r = lower_bound seq

A6: now :: thesis: for s being Real st 0 < s holds
ex r2 being Real st
( r2 in rng seq & r2 < r + s )

let s be Real; :: thesis: ( 0 < s implies ex r2 being Real st
( r2 in rng seq & r2 < r + s ) )
assume 0 < s ; :: thesis: ex r2 being Real st
( r2 in rng seq & r2 < r + s )
then consider k being Nat such that
A7: seq . k < r + s by A5;
A8: k in NAT by ORDINAL1:def 12;
consider r2 being Real such that
A9: ( r2 in rng seq & r2 = seq . k ) by FUNCT_2:4, A8;
take r2 = r2; :: thesis: ( r2 in rng seq & r2 < r + s )
thus ( r2 in rng seq & r2 < r + s ) by A7, A9; :: thesis: verum

end;

now :: thesis: for r1 being Real st r1 in rng seq holds
r <= r1

let r1 be Real; :: thesis: ( r1 in rng seq implies r <= r1 )
assume r1 in rng seq ; :: thesis: r <= r1
then ex n being object st
( n in dom seq & seq . n = r1 ) by FUNCT_1:def 3;
hence r <= r1 by A4; :: thesis: verum

end;

hence r = lower_bound seq by A1, A2, A6, SEQ_4:def 2; :: thesis: verum

end;

( r = lower_bound seq implies ( ( for n being Nat holds r <= seq . n ) & ( for s being Real st 0 < s holds
ex k being Nat st seq . k < r + s ) ) )

proof

assume A10: r = lower_bound seq ; :: thesis: ( ( for n being Nat holds r <= seq . n ) & ( for s being Real st 0 < s holds
ex k being Nat st seq . k < r + s ) )

A11: now :: thesis: for s being Real st 0 < s holds
ex k being Nat st seq . k < r + s

let s be Real; :: thesis: ( 0 < s implies ex k being Nat st seq . k < r + s )
assume 0 < s ; :: thesis: ex k being Nat st seq . k < r + s
then consider r2 being Real such that
A12: r2 in rng seq and
A13: r2 < r + s by A1, A2, A10, SEQ_4:def 2;
consider k being Nat such that
A14: r2 = seq . k by A12, SETLIM_1:4;
reconsider k = k as Nat ;
take k = k; :: thesis: seq . k < r + s
thus seq . k < r + s by A13, A14; :: thesis: verum

end;

now :: thesis: for n being Nat holds r <= seq . n

let n be Nat; :: thesis: r <= seq . n
n in NAT by ORDINAL1:def 12;
then seq . n in rng seq by FUNCT_2:4;
hence r <= seq . n by A1, A10, SEQ_4:def 2; :: thesis: verum

end;

hence ( ( for n being Nat holds r <= seq . n ) & ( for s being Real st 0 < s holds
ex k being Nat st seq . k < r + s ) ) by A11; :: thesis: verum

end;

hence ( r = lower_bound seq iff ( ( for n being Nat holds r <= seq . n ) & ( for s being Real st 0 < s holds
ex k being Nat st seq . k < r + s ) ) ) by A3; :: thesis: verum