{ (q `1) where q is Point of () : ( q `2 = S-bound (L~ g) & q in L~ g ) } c= REAL
proof
let X be object ; :: according to TARSKI:def 3 :: thesis: ( not X in { (q `1) where q is Point of () : ( q `2 = S-bound (L~ g) & q in L~ g ) } or X in REAL )
assume X in { (q `1) where q is Point of () : ( q `2 = S-bound (L~ g) & q in L~ g ) } ; :: thesis:
then ex q being Point of () st
( X = q `1 & q `2 = S-bound (L~ g) & q in L~ g ) ;
hence X in REAL by XREAL_0:def 1; :: thesis: verum
end;
then reconsider B = { (q `1) where q is Point of () : ( q `2 = S-bound (L~ g) & q in L~ g ) } as Subset of REAL ;
set s1 = ((GoB g) * (1,1)) `1 ;
defpred S1[ Nat] means ( [\$1,1] in Indices (GoB g) & ex i being 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 object ; :: 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 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_0:def 4;
hence y in dom (GoB g) by ZFMISC_1:87; :: thesis: verum
end;
A82: { j where j is Element of NAT : S1[j] } is Subset of NAT from A83: 1 <= width (GoB g) by GOBOARD7:33;
then consider i, j being 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 Th8;
j in NAT by ORDINAL1:def 12;
then j in { j where j is Element of NAT : S1[j] } by ;
then reconsider Y = { j where j is Element of NAT : S1[j] } as non empty finite Subset of NAT by ;
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 Nat st
( i in dom g & g /. i = (GoB g) * (j,1) ) ;
A90: min Y <= len (GoB g) by ;
A91: 1 <= width (GoB g) by ;
1 <= min Y by ;
then A92: ((GoB g) * ((min Y),1)) `2 = ((GoB g) * (1,1)) `2 by ;
then A93: ((GoB g) * ((min Y),1)) `2 = S-bound (L~ g) by Th38;
consider i being Nat such that
A94: i in dom g and
A95: g /. i = (GoB g) * (j,1) by A89;
A96: i <= len g by ;
A97: 1 <= i by ;
A98: now :: thesis: ( ( i < len g & (GoB g) * ((min Y),1) in L~ g ) or ( i = len g & (GoB g) * ((min Y),1) in L~ g ) )
per cases ( i < len g or i = len g ) by ;
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 ;
hence (GoB g) * ((min Y),1) in L~ g by ; :: 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 ;
hence (GoB g) * ((min Y),1) in L~ g by ; :: thesis: verum
end;
end;
end;
((GoB g) * ((min Y),1)) `2 = S-bound (L~ g) by ;
then A99: ((GoB g) * ((min Y),1)) `1 in { (q `1) where q is Point of () : ( q `2 = S-bound (L~ g) & q in L~ g ) } by A98;
for r being Real st r in B holds
r >= ((GoB g) * ((min Y),1)) `1
proof
let r be Real; :: 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 () 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 ;
((GoB g) * (1,1)) `1 is LowerBound of B
proof
let r be ExtReal; :: according to XXREAL_2:def 2 :: thesis: ( not r in B or ((GoB g) * (1,1)) `1 <= r )
assume r in B ; :: thesis: ((GoB g) * (1,1)) `1 <= r
then ex q being Point of () st
( r = q `1 & q `2 = S-bound (L~ g) & q in L~ g ) ;
hence ((GoB g) * (1,1)) `1 <= r by ; :: thesis: verum
end;
then B is bounded_below ;
then lower_bound B <= ((GoB g) * ((min Y),1)) `1 by ;
then ((GoB g) * ((min Y),1)) `1 = lower_bound B by
.= lower_bound (proj1 | (S-most (L~ g))) by Th16 ;
hence ex b1 being Nat st
( [b1,1] in Indices (GoB g) & (GoB g) * (b1,1) = S-min (L~ g) ) by ; :: thesis: verum