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

proof

let X be set ; :: according to TARSKI:def 3 :: thesis: ( not X in { (q `2 ) where q is Point of (TOP-REAL 2) : ( q `1 = W-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 = W-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 = W-bound (L~ g) & q in L~ g ) ;
hence X in REAL ; :: thesis: verum

end;

then reconsider B = { (q `2 ) where q is Point of (TOP-REAL 2) : ( q `1 = W-bound (L~ g) & q in L~ g ) } as Subset of REAL ;
set s1 = ((GoB g) * 1,(width (GoB g))) `2 ;
defpred S₁[ Element of NAT ] means ( [1,$1] in Indices (GoB g) & ex i being Element of NAT st
( i in dom g & g /. i = (GoB g) * 1,$1 ) );
set Y = { j where j is Element of NAT : S₁[j] } ;
A21: { j where j is Element of NAT : S₁[j] } c= Seg (width (GoB g))

proof

let y be set ; :: according to TARSKI:def 3 :: thesis: ( not y in { j where j is Element of NAT : S₁[j] } or y in Seg (width (GoB g)) )
assume y in { j where j is Element of NAT : S₁[j] } ; :: thesis: y in Seg (width (GoB g))
then ex j being Element of NAT st
( y = j & [1,j] in Indices (GoB g) & ex i being Element of NAT st
( i in dom g & g /. i = (GoB g) * 1,j ) ) ;
then [1,y] in [:(dom (GoB g)),(Seg (width (GoB g))):] by MATRIX_1:def 5;
hence y in Seg (width (GoB g)) by ZFMISC_1:106; :: thesis: verum

end;

A22: { j where j is Element of NAT : S₁[j] } is Subset of NAT from DOMAIN_1:sch 7();
A23: 1 <= len (GoB g) by GOBOARD7:34;
then consider i, j being Element of NAT such that
A24: i in dom g and
A25: [1,j] in Indices (GoB g) and
A26: g /. i = (GoB g) * 1,j by Th9;
j in { j where j is Element of NAT : S₁[j] } by A24, A25, A26;
then reconsider Y = { j where j is Element of NAT : S₁[j] } as non empty finite Subset of NAT by A21, A22;
reconsider i1 = max Y as Element of NAT by ORDINAL1:def 13;
i1 in Y by XXREAL_2:def 8;
then consider j being Element of NAT such that
A27: j = i1 and
A28: [1,j] in Indices (GoB g) and
A29: ex i being Element of NAT st
( i in dom g & g /. i = (GoB g) * 1,j ) ;
A30: i1 <= width (GoB g) by A27, A28, MATRIX_1:39;
A31: 1 <= len (GoB g) by A28, MATRIX_1:39;
1 <= i1 by A27, A28, MATRIX_1:39;
then A32: ((GoB g) * 1,i1) `1 = ((GoB g) * 1,1) `1 by A31, A30, GOBOARD5:3;
then A33: ((GoB g) * 1,i1) `1 = W-bound (L~ g) by Th39;
consider i being Element of NAT such that
A34: i in dom g and
A35: g /. i = (GoB g) * 1,j by A29;
A36: i <= len g by A34, FINSEQ_3:27;
A37: 1 <= i by A34, FINSEQ_3:27;

A38: now

per cases ( i < len g or i = len g ) by A36, XXREAL_0:1;

case i < len g ; :: thesis: (GoB g) * 1,i1 in L~ g

then i + 1 <= len g by NAT_1:13;
then g /. i in LSeg g,i by A37, TOPREAL1:27;
hence (GoB g) * 1,i1 in L~ g by A27, A35, SPPOL_2:17; :: thesis: verum

end;

case i = len g ; :: thesis: (GoB g) * 1,i1 in L~ g

then g /. i in LSeg g,(i -' 1) by Lm9, Th3;
hence (GoB g) * 1,i1 in L~ g by A27, A35, SPPOL_2:17; :: thesis: verum

end;

end;

end;

((GoB g) * 1,i1) `1 = W-bound (L~ g) by A32, Th39;
then A39: ((GoB g) * 1,i1) `2 in { (q `2 ) where q is Point of (TOP-REAL 2) : ( q `1 = W-bound (L~ g) & q in L~ g ) } by A38;
for r being real number st r in B holds
r <= ((GoB g) * 1,i1) `2

proof

let r be real number ; :: thesis: ( r in B implies r <= ((GoB g) * 1,i1) `2 )
assume r in B ; :: thesis: r <= ((GoB g) * 1,i1) `2
then ex q being Point of (TOP-REAL 2) st
( r = q `2 & q `1 = W-bound (L~ g) & q in L~ g ) ;
hence r <= ((GoB g) * 1,i1) `2 by Lm2; :: thesis: verum

end;

then A40: upper_bound B <= ((GoB g) * 1,i1) `2 by A39, SEQ_4:62;
for r being real number st r in B holds
((GoB g) * 1,(width (GoB g))) `2 >= r

proof

let r be real number ; :: thesis: ( r in B implies ((GoB g) * 1,(width (GoB g))) `2 >= r )
assume r in B ; :: thesis: ((GoB g) * 1,(width (GoB g))) `2 >= r
then ex q being Point of (TOP-REAL 2) st
( r = q `2 & q `1 = W-bound (L~ g) & q in L~ g ) ;
hence ((GoB g) * 1,(width (GoB g))) `2 >= r by A23, Th36; :: thesis: verum

end;

then B is bounded_above by SEQ_4:def 1;
then upper_bound B >= ((GoB g) * 1,i1) `2 by A39, SEQ_4:def 4;
then ((GoB g) * 1,i1) `2 = upper_bound B by A40, XXREAL_0:1
.= sup (proj2 | (W-most (L~ g))) by Th15 ;
hence ex b₁ being Element of NAT st
( [1,b₁] in Indices (GoB g) & (GoB g) * 1,b₁ = W-max (L~ g) ) by A27, A28, A33, EUCLID:57; :: thesis: verum