{ (q `1) where q is Point of (TOP-REAL 2) : ( q `2 = N-bound (L~ g) & q in L~ g ) } c= REAL

defpred S_{1}[ Nat] means ( [$1,(width (GoB g))] in Indices (GoB g) & ex i being Nat st

( i in dom g & g /. i = (GoB g) * ($1,(width (GoB g))) ) );

set Y = { j where j is Element of NAT : S_{1}[j] } ;

A121: { j where j is Element of NAT : S_{1}[j] } c= dom (GoB g)
_{1}[j] } is Subset of NAT
from DOMAIN_1:sch 7();

1 <= width (GoB g) by GOBOARD7:33;

then consider i, j being Nat such that

A123: i in dom g and

A124: [j,(width (GoB g))] in Indices (GoB g) and

A125: g /. i = (GoB g) * (j,(width (GoB g))) by Th8;

j in NAT by ORDINAL1:def 12;

then j in { j where j is Element of NAT : S_{1}[j] }
by A123, A124, A125;

then reconsider Y = { j where j is Element of NAT : S_{1}[j] } as non empty finite Subset of NAT by A121, A122;

set i1 = min Y;

set s1 = ((GoB g) * (1,(width (GoB g)))) `1 ;

min Y in Y by XXREAL_2:def 7;

then consider j being Element of NAT such that

A126: j = min Y and

A127: [j,(width (GoB g))] in Indices (GoB g) and

A128: ex i being Nat st

( i in dom g & g /. i = (GoB g) * (j,(width (GoB g))) ) ;

A129: min Y <= len (GoB g) by A126, A127, MATRIX_0:32;

A130: 1 <= width (GoB g) by A127, MATRIX_0:32;

1 <= min Y by A126, A127, MATRIX_0:32;

then A131: ((GoB g) * ((min Y),(width (GoB g)))) `2 = ((GoB g) * (1,(width (GoB g)))) `2 by A129, A130, GOBOARD5:1;

then A132: ((GoB g) * ((min Y),(width (GoB g)))) `2 = N-bound (L~ g) by Th40;

consider i being Nat such that

A133: i in dom g and

A134: g /. i = (GoB g) * (j,(width (GoB g))) by A128;

A135: i <= len g by A133, FINSEQ_3:25;

A136: 1 <= i by A133, FINSEQ_3:25;

then A138: ((GoB g) * ((min Y),(width (GoB g)))) `1 in { (q `1) where q is Point of (TOP-REAL 2) : ( q `2 = N-bound (L~ g) & q in L~ g ) } by A137;

for r being Real st r in B holds

r >= ((GoB g) * ((min Y),(width (GoB g)))) `1

((GoB g) * (1,(width (GoB g)))) `1 is LowerBound of B

then lower_bound B <= ((GoB g) * ((min Y),(width (GoB g)))) `1 by A138, SEQ_4:def 2;

then ((GoB g) * ((min Y),(width (GoB g)))) `1 = lower_bound B by A139, XXREAL_0:1

.= lower_bound (proj1 | (N-most (L~ g))) by Th15 ;

hence ex b_{1} being Nat st

( [b_{1},(width (GoB g))] in Indices (GoB g) & (GoB g) * (b_{1},(width (GoB g))) = N-min (L~ g) )
by A126, A127, A132, EUCLID:53; :: thesis: verum

proof

then reconsider B = { (q `1) where q is Point of (TOP-REAL 2) : ( q `2 = N-bound (L~ g) & q in L~ g ) } as Subset of REAL ;
let X be object ; :: according to TARSKI:def 3 :: thesis: ( not X in { (q `1) where q is Point of (TOP-REAL 2) : ( q `2 = N-bound (L~ g) & q in L~ g ) } or X in REAL )

assume X in { (q `1) where q is Point of (TOP-REAL 2) : ( q `2 = N-bound (L~ g) & q in L~ g ) } ; :: thesis: X in REAL

then ex q being Point of (TOP-REAL 2) st

( X = q `1 & q `2 = N-bound (L~ g) & q in L~ g ) ;

hence X in REAL by XREAL_0:def 1; :: thesis: verum

end;assume X in { (q `1) where q is Point of (TOP-REAL 2) : ( q `2 = N-bound (L~ g) & q in L~ g ) } ; :: thesis: X in REAL

then ex q being Point of (TOP-REAL 2) st

( X = q `1 & q `2 = N-bound (L~ g) & q in L~ g ) ;

hence X in REAL by XREAL_0:def 1; :: thesis: verum

defpred S

( i in dom g & g /. i = (GoB g) * ($1,(width (GoB g))) ) );

set Y = { j where j is Element of NAT : S

A121: { j where j is Element of NAT : S

proof

A122:
{ j where j is Element of NAT : S
let y be object ; :: according to TARSKI:def 3 :: thesis: ( not y in { j where j is Element of NAT : S_{1}[j] } or y in dom (GoB g) )

assume y in { j where j is Element of NAT : S_{1}[j] }
; :: thesis: y in dom (GoB g)

then ex j being Element of NAT st

( y = j & [j,(width (GoB g))] in Indices (GoB g) & ex i being Nat st

( i in dom g & g /. i = (GoB g) * (j,(width (GoB g))) ) ) ;

then [y,(width (GoB g))] in [:(dom (GoB g)),(Seg (width (GoB g))):] by MATRIX_0:def 4;

hence y in dom (GoB g) by ZFMISC_1:87; :: thesis: verum

end;assume y in { j where j is Element of NAT : S

then ex j being Element of NAT st

( y = j & [j,(width (GoB g))] in Indices (GoB g) & ex i being Nat st

( i in dom g & g /. i = (GoB g) * (j,(width (GoB g))) ) ) ;

then [y,(width (GoB g))] in [:(dom (GoB g)),(Seg (width (GoB g))):] by MATRIX_0:def 4;

hence y in dom (GoB g) by ZFMISC_1:87; :: thesis: verum

1 <= width (GoB g) by GOBOARD7:33;

then consider i, j being Nat such that

A123: i in dom g and

A124: [j,(width (GoB g))] in Indices (GoB g) and

A125: g /. i = (GoB g) * (j,(width (GoB g))) by Th8;

j in NAT by ORDINAL1:def 12;

then j in { j where j is Element of NAT : S

then reconsider Y = { j where j is Element of NAT : S

set i1 = min Y;

set s1 = ((GoB g) * (1,(width (GoB g)))) `1 ;

min Y in Y by XXREAL_2:def 7;

then consider j being Element of NAT such that

A126: j = min Y and

A127: [j,(width (GoB g))] in Indices (GoB g) and

A128: ex i being Nat st

( i in dom g & g /. i = (GoB g) * (j,(width (GoB g))) ) ;

A129: min Y <= len (GoB g) by A126, A127, MATRIX_0:32;

A130: 1 <= width (GoB g) by A127, MATRIX_0:32;

1 <= min Y by A126, A127, MATRIX_0:32;

then A131: ((GoB g) * ((min Y),(width (GoB g)))) `2 = ((GoB g) * (1,(width (GoB g)))) `2 by A129, A130, GOBOARD5:1;

then A132: ((GoB g) * ((min Y),(width (GoB g)))) `2 = N-bound (L~ g) by Th40;

consider i being Nat such that

A133: i in dom g and

A134: g /. i = (GoB g) * (j,(width (GoB g))) by A128;

A135: i <= len g by A133, FINSEQ_3:25;

A136: 1 <= i by A133, FINSEQ_3:25;

A137: now :: thesis: ( ( i < len g & (GoB g) * ((min Y),(width (GoB g))) in L~ g ) or ( i = len g & (GoB g) * ((min Y),(width (GoB g))) in L~ g ) )end;

((GoB g) * ((min Y),(width (GoB g)))) `2 = N-bound (L~ g)
by A131, Th40;per cases
( i < len g or i = len g )
by A135, XXREAL_0:1;

end;

then A138: ((GoB g) * ((min Y),(width (GoB g)))) `1 in { (q `1) where q is Point of (TOP-REAL 2) : ( q `2 = N-bound (L~ g) & q in L~ g ) } by A137;

for r being Real st r in B holds

r >= ((GoB g) * ((min Y),(width (GoB g)))) `1

proof

then A139:
lower_bound B >= ((GoB g) * ((min Y),(width (GoB g)))) `1
by A138, SEQ_4:43;
let r be Real; :: thesis: ( r in B implies r >= ((GoB g) * ((min Y),(width (GoB g)))) `1 )

assume r in B ; :: thesis: r >= ((GoB g) * ((min Y),(width (GoB g)))) `1

then ex q being Point of (TOP-REAL 2) st

( r = q `1 & q `2 = N-bound (L~ g) & q in L~ g ) ;

hence r >= ((GoB g) * ((min Y),(width (GoB g)))) `1 by Lm6; :: thesis: verum

end;assume r in B ; :: thesis: r >= ((GoB g) * ((min Y),(width (GoB g)))) `1

then ex q being Point of (TOP-REAL 2) st

( r = q `1 & q `2 = N-bound (L~ g) & q in L~ g ) ;

hence r >= ((GoB g) * ((min Y),(width (GoB g)))) `1 by Lm6; :: thesis: verum

((GoB g) * (1,(width (GoB g)))) `1 is LowerBound of B

proof

then
B is bounded_below
;
let r be ExtReal; :: according to XXREAL_2:def 2 :: thesis: ( not r in B or ((GoB g) * (1,(width (GoB g)))) `1 <= r )

assume r in B ; :: thesis: ((GoB g) * (1,(width (GoB g)))) `1 <= r

then A140: ex q1 being Point of (TOP-REAL 2) st

( r = q1 `1 & q1 `2 = N-bound (L~ g) & q1 in L~ g ) ;

1 <= width (GoB g) by GOBOARD7:33;

hence ((GoB g) * (1,(width (GoB g)))) `1 <= r by A140, Th31; :: thesis: verum

end;assume r in B ; :: thesis: ((GoB g) * (1,(width (GoB g)))) `1 <= r

then A140: ex q1 being Point of (TOP-REAL 2) st

( r = q1 `1 & q1 `2 = N-bound (L~ g) & q1 in L~ g ) ;

1 <= width (GoB g) by GOBOARD7:33;

hence ((GoB g) * (1,(width (GoB g)))) `1 <= r by A140, Th31; :: thesis: verum

then lower_bound B <= ((GoB g) * ((min Y),(width (GoB g)))) `1 by A138, SEQ_4:def 2;

then ((GoB g) * ((min Y),(width (GoB g)))) `1 = lower_bound B by A139, XXREAL_0:1

.= lower_bound (proj1 | (N-most (L~ g))) by Th15 ;

hence ex b

( [b