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

proof

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

end;

then reconsider B = { (q `1 ) where q is Point of (TOP-REAL 2) : ( q `2 = S-bound (L~ g) & q in L~ g ) } as Subset of REAL ;
set s1 = ((GoB g) * 1,1) `1 ;
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] } ;
A81: { j where j is Element of NAT : S₁[j] } c= dom (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 dom (GoB g) )
assume y in { j where j is Element of NAT : S₁[j] } ; :: thesis: y in dom (GoB g)
then ex j being Element of NAT st
( y = j & [j,1] in Indices (GoB g) & ex i being Element of NAT st
( i in dom g & g /. i = (GoB g) * j,1 ) ) ;
then [y,1] in [:(dom (GoB g)),(Seg (width (GoB g))):] by MATRIX_1:def 5;
hence y in dom (GoB g) by ZFMISC_1:106; :: thesis: verum

end;

A82: { j where j is Element of NAT : S₁[j] } is Subset of NAT from DOMAIN_1:sch 7();
A83: 1 <= width (GoB g) by GOBOARD7:35;
then consider i, j being Element of NAT such that
A84: i in dom g and
A85: [j,1] in Indices (GoB g) and
A86: g /. i = (GoB g) * j,1 by Th10;
j in { j where j is Element of NAT : S₁[j] } by A84, A85, A86;
then reconsider Y = { j where j is Element of NAT : S₁[j] } as non empty finite Subset of NAT by A81, A82;
set i1 = min Y;
min Y in Y by XXREAL_2:def 7;
then consider j being Element of NAT such that
A87: j = min Y and
A88: [j,1] in Indices (GoB g) and
A89: ex i being Element of NAT st
( i in dom g & g /. i = (GoB g) * j,1 ) ;
A90: min Y <= len (GoB g) by A87, A88, MATRIX_1:39;
A91: 1 <= width (GoB g) by A88, MATRIX_1:39;
1 <= min Y by A87, A88, MATRIX_1:39;
then A92: ((GoB g) * (min Y),1) `2 = ((GoB g) * 1,1) `2 by A90, A91, GOBOARD5:2;
then A93: ((GoB g) * (min Y),1) `2 = S-bound (L~ g) by Th40;
consider i being Element of NAT such that
A94: i in dom g and
A95: g /. i = (GoB g) * j,1 by A89;
A96: i <= len g by A94, FINSEQ_3:27;
A97: 1 <= i by A94, FINSEQ_3:27;

A98: now

per cases ( i < len g or i = len g ) by A96, XXREAL_0:1;

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

then i + 1 <= len g by NAT_1:13;
then g /. i in LSeg g,i by A97, TOPREAL1:27;
hence (GoB g) * (min Y),1 in L~ g by A87, A95, SPPOL_2:17; :: thesis: verum

end;

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

then g /. i in LSeg g,(i -' 1) by Lm9, Th3;
hence (GoB g) * (min Y),1 in L~ g by A87, A95, SPPOL_2:17; :: thesis: verum

end;

end;

end;

((GoB g) * (min Y),1) `2 = S-bound (L~ g) by A92, Th40;
then A99: ((GoB g) * (min Y),1) `1 in { (q `1 ) where q is Point of (TOP-REAL 2) : ( q `2 = S-bound (L~ g) & q in L~ g ) } by A98;
for r being real number st r in B holds
r >= ((GoB g) * (min Y),1) `1

proof

let r be real number ; :: thesis: ( r in B implies r >= ((GoB g) * (min Y),1) `1 )
assume r in B ; :: thesis: r >= ((GoB g) * (min Y),1) `1
then ex q being Point of (TOP-REAL 2) st
( r = q `1 & q `2 = S-bound (L~ g) & q in L~ g ) ;
hence r >= ((GoB g) * (min Y),1) `1 by Lm5; :: thesis: verum

end;

then A100: lower_bound B >= ((GoB g) * (min Y),1) `1 by A99, SEQ_4:60;
for r being real number st r in B holds
((GoB g) * 1,1) `1 <= r

proof

let r be real number ; :: thesis: ( r in B implies ((GoB g) * 1,1) `1 <= r )
assume r in B ; :: thesis: ((GoB g) * 1,1) `1 <= r
then ex q being Point of (TOP-REAL 2) st
( r = q `1 & q `2 = S-bound (L~ g) & q in L~ g ) ;
hence ((GoB g) * 1,1) `1 <= r by A83, Th33; :: thesis: verum

end;

then B is bounded_below by SEQ_4:def 2;
then lower_bound B <= ((GoB g) * (min Y),1) `1 by A99, SEQ_4:def 5;
then ((GoB g) * (min Y),1) `1 = lower_bound B by A100, XXREAL_0:1
.= inf (proj1 | (S-most (L~ g))) by Th18 ;
hence ex b₁ being Element of NAT st
( [b₁,1] in Indices (GoB g) & (GoB g) * b₁,1 = S-min (L~ g) ) by A87, A88, A93, EUCLID:57; :: thesis: verum