{ (q `2 ) where q is Point of (TOP-REAL 2) : ( q `1 = E-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 = 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 ; :: thesis: verum

end;

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 ;
set s1 = ((GoB g) * (len (GoB g)),1) `2 ;
defpred S₁[ Element of NAT ] means ( [(len (GoB g)),$1] in Indices (GoB g) & ex i being Element of NAT st
( i in dom g & g /. i = (GoB g) * (len (GoB g)),$1 ) );
set Y = { j where j is Element of NAT : S₁[j] } ;
A41: { 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 & [(len (GoB g)),j] in Indices (GoB g) & ex i being Element of 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_1:def 5;
hence y in Seg (width (GoB g)) by ZFMISC_1:106; :: thesis: verum

end;

A42: { j where j is Element of NAT : S₁[j] } is Subset of NAT from DOMAIN_1:sch 7();
A43: 1 <= len (GoB g) by GOBOARD7:34;
then consider i, j being Element of NAT such that
A44: i in dom g and
A45: [(len (GoB g)),j] in Indices (GoB g) and
A46: g /. i = (GoB g) * (len (GoB g)),j by Th9;
j in { j where j is Element of NAT : S₁[j] } by A44, A45, A46;
then reconsider Y = { j where j is Element of NAT : S₁[j] } as non empty finite Subset of NAT by A41, A42;
set i1 = min Y;
min Y in Y by XXREAL_2:def 7;
then consider j being Element of NAT such that
A47: j = min Y and
A48: [(len (GoB g)),j] in Indices (GoB g) and
A49: ex i being Element of NAT st
( i in dom g & g /. i = (GoB g) * (len (GoB g)),j ) ;
A50: min Y <= width (GoB g) by A47, A48, MATRIX_1:39;
A51: 1 <= len (GoB g) by A48, MATRIX_1:39;
1 <= min Y by A47, A48, MATRIX_1:39;
then A52: ((GoB g) * (len (GoB g)),(min Y)) `1 = ((GoB g) * (len (GoB g)),1) `1 by A51, A50, GOBOARD5:3;
then A53: ((GoB g) * (len (GoB g)),(min Y)) `1 = E-bound (L~ g) by Th41;
consider i being Element of NAT such that
A54: i in dom g and
A55: g /. i = (GoB g) * (len (GoB g)),j by A49;
A56: i <= len g by A54, FINSEQ_3:27;
A57: 1 <= i by A54, FINSEQ_3:27;

A58: now

per cases ( i < len g or i = len g ) by A56, XXREAL_0:1;

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

then i + 1 <= len g by NAT_1:13;
then g /. i in LSeg g,i by A57, TOPREAL1:27;
hence (GoB g) * (len (GoB g)),(min Y) in L~ g by A47, A55, SPPOL_2:17; :: thesis: verum

end;

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

then g /. i in LSeg g,(i -' 1) by Lm9, Th3;
hence (GoB g) * (len (GoB g)),(min Y) in L~ g by A47, A55, SPPOL_2:17; :: thesis: verum

end;

end;

end;

((GoB g) * (len (GoB g)),(min Y)) `1 = E-bound (L~ g) by A52, Th41;
then A59: ((GoB g) * (len (GoB g)),(min Y)) `2 in { (q `2 ) where q is Point of (TOP-REAL 2) : ( q `1 = E-bound (L~ g) & q in L~ g ) } by A58;
for r being real number st r in B holds
r >= ((GoB g) * (len (GoB g)),(min Y)) `2

proof

let r be real number ; :: thesis: ( r in B implies r >= ((GoB g) * (len (GoB g)),(min Y)) `2 )
assume r in B ; :: thesis: r >= ((GoB g) * (len (GoB g)),(min Y)) `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)),(min Y)) `2 by Lm3; :: thesis: verum

end;

then A60: lower_bound B >= ((GoB g) * (len (GoB g)),(min Y)) `2 by A59, SEQ_4:60;
for r being real number st r in B holds
((GoB g) * (len (GoB g)),1) `2 <= r

proof

let r be real number ; :: thesis: ( r in B implies ((GoB g) * (len (GoB g)),1) `2 <= r )
assume r in B ; :: thesis: ((GoB g) * (len (GoB g)),1) `2 <= r
then ex q being Point of (TOP-REAL 2) st
( r = q `2 & q `1 = E-bound (L~ g) & q in L~ g ) ;
hence ((GoB g) * (len (GoB g)),1) `2 <= r by A43, Th35; :: thesis: verum

end;

then B is bounded_below by SEQ_4:def 2;
then lower_bound B <= ((GoB g) * (len (GoB g)),(min Y)) `2 by A59, SEQ_4:def 5;
then ((GoB g) * (len (GoB g)),(min Y)) `2 = lower_bound B by A60, XXREAL_0:1
.= inf (proj2 | (E-most (L~ g))) by Th16 ;
hence ex b₁ being Element of NAT st
( [(len (GoB g)),b₁] in Indices (GoB g) & (GoB g) * (len (GoB g)),b₁ = E-min (L~ g) ) by A47, A48, A53, EUCLID:57; :: thesis: verum