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

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

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

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

A61: { j where j is Element of NAT : S_{1}[j] } c= Seg (width (GoB g))
_{1}[j] } is Subset of NAT
from DOMAIN_1:sch 7();

0 <> len (GoB g) by MATRIX_0:def 10;

then 1 <= len (GoB g) by NAT_1:14;

then consider i, j being Nat such that

A63: i in dom g and

A64: [(len (GoB g)),j] in Indices (GoB g) and

A65: g /. i = (GoB g) * ((len (GoB g)),j) by Th7;

j in NAT by ORDINAL1:def 12;

then j in { j where j is Element of NAT : S_{1}[j] }
by A63, A64, A65;

then reconsider Y = { j where j is Element of NAT : S_{1}[j] } as non empty finite Subset of NAT by A61, A62;

reconsider i1 = max Y as Nat by TARSKI:1;

set s1 = ((GoB g) * ((len (GoB g)),(width (GoB g)))) `2 ;

i1 in Y by XXREAL_2:def 8;

then consider j being Element of NAT such that

A66: j = i1 and

A67: [(len (GoB g)),j] in Indices (GoB g) and

A68: ex i being Nat st

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

A69: i1 <= width (GoB g) by A66, A67, MATRIX_0:32;

A70: 1 <= len (GoB g) by A67, MATRIX_0:32;

1 <= i1 by A66, A67, MATRIX_0:32;

then A71: ((GoB g) * ((len (GoB g)),i1)) `1 = ((GoB g) * ((len (GoB g)),1)) `1 by A70, A69, GOBOARD5:2;

then A72: ((GoB g) * ((len (GoB g)),i1)) `1 = E-bound (L~ g) by Th39;

consider i being Nat such that

A73: i in dom g and

A74: g /. i = (GoB g) * ((len (GoB g)),j) by A68;

A75: i <= len g by A73, FINSEQ_3:25;

A76: 1 <= i by A73, FINSEQ_3:25;

then A78: ((GoB g) * ((len (GoB g)),i1)) `2 in { (q `2) where q is Point of (TOP-REAL 2) : ( q `1 = E-bound (L~ g) & q in L~ g ) } by A77;

for r being Real st r in B holds

r <= ((GoB g) * ((len (GoB g)),i1)) `2

((GoB g) * ((len (GoB g)),(width (GoB g)))) `2 is UpperBound of B

then upper_bound B >= ((GoB g) * ((len (GoB g)),i1)) `2 by A78, SEQ_4:def 1;

then ((GoB g) * ((len (GoB g)),i1)) `2 = upper_bound B by A79, XXREAL_0:1

.= upper_bound (proj2 | (E-most (L~ g))) by Th14 ;

hence ex b_{1} being Nat st

( [(len (GoB g)),b_{1}] in Indices (GoB g) & (GoB g) * ((len (GoB g)),b_{1}) = E-max (L~ g) )
by A66, A67, A72, EUCLID:53; :: thesis: verum

proof

then reconsider B = { (q `2) where q is Point of (TOP-REAL 2) : ( q `1 = E-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 `2) where q is Point of (TOP-REAL 2) : ( q `1 = E-bound (L~ g) & q in L~ g ) } or X in REAL )

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

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

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

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

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

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

( X = q `2 & q `1 = E-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) * ((len (GoB g)),$1) ) );

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

A61: { j where j is Element of NAT : S

proof

A62:
{ 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 Seg (width (GoB g)) )

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

then ex j being Element of NAT st

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

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

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

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

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

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

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

0 <> len (GoB g) by MATRIX_0:def 10;

then 1 <= len (GoB g) by NAT_1:14;

then consider i, j being Nat such that

A63: i in dom g and

A64: [(len (GoB g)),j] in Indices (GoB g) and

A65: g /. i = (GoB g) * ((len (GoB g)),j) by Th7;

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

reconsider i1 = max Y as Nat by TARSKI:1;

set s1 = ((GoB g) * ((len (GoB g)),(width (GoB g)))) `2 ;

i1 in Y by XXREAL_2:def 8;

then consider j being Element of NAT such that

A66: j = i1 and

A67: [(len (GoB g)),j] in Indices (GoB g) and

A68: ex i being Nat st

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

A69: i1 <= width (GoB g) by A66, A67, MATRIX_0:32;

A70: 1 <= len (GoB g) by A67, MATRIX_0:32;

1 <= i1 by A66, A67, MATRIX_0:32;

then A71: ((GoB g) * ((len (GoB g)),i1)) `1 = ((GoB g) * ((len (GoB g)),1)) `1 by A70, A69, GOBOARD5:2;

then A72: ((GoB g) * ((len (GoB g)),i1)) `1 = E-bound (L~ g) by Th39;

consider i being Nat such that

A73: i in dom g and

A74: g /. i = (GoB g) * ((len (GoB g)),j) by A68;

A75: i <= len g by A73, FINSEQ_3:25;

A76: 1 <= i by A73, FINSEQ_3:25;

A77: now :: thesis: ( ( i < len g & (GoB g) * ((len (GoB g)),i1) in L~ g ) or ( i = len g & (GoB g) * ((len (GoB g)),i1) in L~ g ) )

((GoB g) * ((len (GoB g)),i1)) `1 = E-bound (L~ g)
by A71, Th39;end;

then A78: ((GoB g) * ((len (GoB g)),i1)) `2 in { (q `2) where q is Point of (TOP-REAL 2) : ( q `1 = E-bound (L~ g) & q in L~ g ) } by A77;

for r being Real st r in B holds

r <= ((GoB g) * ((len (GoB g)),i1)) `2

proof

then A79:
upper_bound B <= ((GoB g) * ((len (GoB g)),i1)) `2
by A78, SEQ_4:45;
let r be Real; :: thesis: ( r in B implies r <= ((GoB g) * ((len (GoB g)),i1)) `2 )

assume r in B ; :: thesis: r <= ((GoB g) * ((len (GoB g)),i1)) `2

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

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

hence r <= ((GoB g) * ((len (GoB g)),i1)) `2 by Lm4; :: thesis: verum

end;assume r in B ; :: thesis: r <= ((GoB g) * ((len (GoB g)),i1)) `2

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

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

hence r <= ((GoB g) * ((len (GoB g)),i1)) `2 by Lm4; :: thesis: verum

((GoB g) * ((len (GoB g)),(width (GoB g)))) `2 is UpperBound of B

proof

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

assume r in B ; :: thesis: r <= ((GoB g) * ((len (GoB g)),(width (GoB g)))) `2

then A80: ex q being Point of (TOP-REAL 2) st

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

1 <= len (GoB g) by GOBOARD7:32;

hence r <= ((GoB g) * ((len (GoB g)),(width (GoB g)))) `2 by A80, Th34; :: thesis: verum

end;assume r in B ; :: thesis: r <= ((GoB g) * ((len (GoB g)),(width (GoB g)))) `2

then A80: ex q being Point of (TOP-REAL 2) st

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

1 <= len (GoB g) by GOBOARD7:32;

hence r <= ((GoB g) * ((len (GoB g)),(width (GoB g)))) `2 by A80, Th34; :: thesis: verum

then upper_bound B >= ((GoB g) * ((len (GoB g)),i1)) `2 by A78, SEQ_4:def 1;

then ((GoB g) * ((len (GoB g)),i1)) `2 = upper_bound B by A79, XXREAL_0:1

.= upper_bound (proj2 | (E-most (L~ g))) by Th14 ;

hence ex b

( [(len (GoB g)),b