{ (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) * 1,1) `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] } ;
A1: { 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;

A2: { j where j is Element of NAT : S₁[j] } is Subset of NAT from DOMAIN_1:sch 7();
A3: 1 <= len (GoB g) by GOBOARD7:34;
then consider i, j being Element of NAT such that
A4: i in dom g and
A5: [1,j] in Indices (GoB g) and
A6: g /. i = (GoB g) * 1,j by Th9;
j in { j where j is Element of NAT : S₁[j] } by A4, A5, A6;
then reconsider Y = { j where j is Element of NAT : S₁[j] } as non empty finite Subset of NAT by A1, A2;
set i1 = min Y;
min Y in Y by XXREAL_2:def 7;
then consider j being Element of NAT such that
A7: j = min Y and
A8: [1,j] in Indices (GoB g) and
A9: ex i being Element of NAT st
( i in dom g & g /. i = (GoB g) * 1,j ) ;
A10: min Y <= width (GoB g) by A7, A8, MATRIX_1:39;
A11: 1 <= len (GoB g) by A8, MATRIX_1:39;
1 <= min Y by A7, A8, MATRIX_1:39;
then A12: ((GoB g) * 1,(min Y)) `1 = ((GoB g) * 1,1) `1 by A11, A10, GOBOARD5:3;
then A13: ((GoB g) * 1,(min Y)) `1 = W-bound (L~ g) by Th39;
consider i being Element of NAT such that
A14: i in dom g and
A15: g /. i = (GoB g) * 1,j by A9;
A16: i <= len g by A14, FINSEQ_3:27;
A17: 1 <= i by A14, FINSEQ_3:27;

A18: now

per cases ( i < len g or i = len g ) by A16, XXREAL_0:1;

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

then i + 1 <= len g by NAT_1:13;
then g /. i in LSeg g,i by A17, TOPREAL1:27;
hence (GoB g) * 1,(min Y) in L~ g by A7, A15, SPPOL_2:17; :: thesis: verum

end;

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

then g /. i in LSeg g,(i -' 1) by Lm9, Th3;
hence (GoB g) * 1,(min Y) in L~ g by A7, A15, SPPOL_2:17; :: thesis: verum

end;

end;

end;

((GoB g) * 1,(min Y)) `1 = W-bound (L~ g) by A12, Th39;
then A19: ((GoB g) * 1,(min Y)) `2 in { (q `2 ) where q is Point of (TOP-REAL 2) : ( q `1 = W-bound (L~ g) & q in L~ g ) } by A18;
for r being real number st r in B holds
r >= ((GoB g) * 1,(min Y)) `2

proof

let r be real number ; :: thesis: ( r in B implies r >= ((GoB g) * 1,(min Y)) `2 )
assume r in B ; :: thesis: r >= ((GoB g) * 1,(min Y)) `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,(min Y)) `2 by Lm1; :: thesis: verum

end;

then A20: lower_bound B >= ((GoB g) * 1,(min Y)) `2 by A19, SEQ_4:60;
((GoB g) * 1,1) `2 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) * 1,1) `2 <= r )
assume r in B ; :: thesis: ((GoB g) * 1,1) `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) * 1,1) `2 <= r by A3, Th35; :: thesis: verum

end;

then B is bounded_below by XXREAL_2:def 9;
then lower_bound B <= ((GoB g) * 1,(min Y)) `2 by A19, SEQ_4:def 5;
then ((GoB g) * 1,(min Y)) `2 = lower_bound B by A20, XXREAL_0:1
.= lower_bound (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-min (L~ g) ) by A7, A8, A13, EUCLID:57; :: thesis: verum