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

end;

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 ;
defpred S₁[ Element of NAT ] means ( [$1,(width (GoB g))] in Indices (GoB g) & ex i being Element of NAT st
( i in dom g & g /. i = (GoB g) * $1,(width (GoB g)) ) );
set Y = { j where j is Element of NAT : S₁[j] } ;
A121: { 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,(width (GoB g))] in Indices (GoB g) & ex i being Element of 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_1:def 5;
hence y in dom (GoB g) by ZFMISC_1:106; :: thesis: verum

end;

A122: { j where j is Element of NAT : S₁[j] } is Subset of NAT from DOMAIN_1:sch 7();
1 <= width (GoB g) by GOBOARD7:35;
then consider i, j being Element of 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 Th10;
j in { j where j is Element of NAT : S₁[j] } by A123, A124, A125;
then reconsider Y = { j where j is Element of NAT : S₁[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 Element of NAT st
( i in dom g & g /. i = (GoB g) * j,(width (GoB g)) ) ;
A129: min Y <= len (GoB g) by A126, A127, MATRIX_1:39;
A130: 1 <= width (GoB g) by A127, MATRIX_1:39;
1 <= min Y by A126, A127, MATRIX_1:39;
then A131: ((GoB g) * (min Y),(width (GoB g))) `2 = ((GoB g) * 1,(width (GoB g))) `2 by A129, A130, GOBOARD5:2;
then A132: ((GoB g) * (min Y),(width (GoB g))) `2 = N-bound (L~ g) by Th42;
consider i being Element of 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_3:27;
A136: 1 <= i by A133, FINSEQ_3:27;

A137: now

per cases ( i < len g or i = len g ) by A135, XXREAL_0:1;

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

then i + 1 <= len g by NAT_1:13;
then g /. i in LSeg g,i by A136, TOPREAL1:27;
hence (GoB g) * (min Y),(width (GoB g)) in L~ g by A126, A134, SPPOL_2:17; :: thesis: verum

end;

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

then g /. i in LSeg g,(i -' 1) by Lm9, Th3;
hence (GoB g) * (min Y),(width (GoB g)) in L~ g by A126, A134, SPPOL_2:17; :: thesis: verum

end;

end;

end;

((GoB g) * (min Y),(width (GoB g))) `2 = N-bound (L~ g) by A131, Th42;
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 number st r in B holds
r >= ((GoB g) * (min Y),(width (GoB g))) `1

proof

let r be real number ; :: 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;

then A139: lower_bound B >= ((GoB g) * (min Y),(width (GoB g))) `1 by A138, SEQ_4:60;
((GoB g) * 1,(width (GoB g))) `1 is LowerBound of B

proof

let r be ext-real number ; :: 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 GOBOARD7:35;
hence ((GoB g) * 1,(width (GoB g))) `1 <= r by A140, Th33; :: thesis: verum

end;

then B is bounded_below by XXREAL_2:def 9;
then lower_bound B <= ((GoB g) * (min Y),(width (GoB g))) `1 by A138, SEQ_4:def 5;
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 Th17 ;
hence ex b₁ being Element of NAT st
( [b₁,(width (GoB g))] in Indices (GoB g) & (GoB g) * b₁,(width (GoB g)) = N-min (L~ g) ) by A126, A127, A132, EUCLID:57; :: thesis: verum