{ (q `1) where q is Point of () : ( q `2 = N-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 = N-bound (L~ g) & q in L~ g ) } or X in REAL )
assume X in { (q `1) where q is Point of () : ( q `2 = N-bound (L~ g) & q in L~ g ) } ; :: thesis:
then ex q being Point of () st
( X = q `1 & q `2 = N-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 = N-bound (L~ g) & q in L~ g ) } as Subset of REAL ;
defpred S1[ Nat] means ( [\$1,(width (GoB g))] in Indices (GoB g) & ex i being 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] } ;
A141: { 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,(width (GoB g))] in Indices (GoB g) & ex i being 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_0:def 4;
hence y in dom (GoB g) by ZFMISC_1:87; :: thesis: verum
end;
A142: { j where j is Element of NAT : S1[j] } is Subset of NAT from 1 <= width (GoB g) by GOBOARD7:33;
then consider i, j being Nat such that
A143: i in dom g and
A144: [j,(width (GoB g))] in Indices (GoB g) and
A145: g /. i = (GoB g) * (j,(width (GoB g))) 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 ;
reconsider i1 = max Y as Nat by TARSKI:1;
set s1 = ((GoB g) * ((len (GoB g)),(width (GoB g)))) `1 ;
i1 in Y by XXREAL_2:def 8;
then consider j being Element of NAT such that
A146: j = i1 and
A147: [j,(width (GoB g))] in Indices (GoB g) and
A148: ex i being Nat st
( i in dom g & g /. i = (GoB g) * (j,(width (GoB g))) ) ;
A149: i1 <= len (GoB g) by ;
A150: 1 <= width (GoB g) by ;
1 <= i1 by ;
then A151: ((GoB g) * (i1,(width (GoB g)))) `2 = ((GoB g) * (1,(width (GoB g)))) `2 by ;
then A152: ((GoB g) * (i1,(width (GoB g)))) `2 = N-bound (L~ g) by Th40;
consider i being Nat such that
A153: i in dom g and
A154: g /. i = (GoB g) * (j,(width (GoB g))) by A148;
A155: i <= len g by ;
A156: 1 <= i by ;
A157: now :: thesis: ( ( i < len g & (GoB g) * (i1,(width (GoB g))) in L~ g ) or ( i = len g & (GoB g) * (i1,(width (GoB g))) in L~ g ) )
per cases ( i < len g or i = len g ) by ;
case i < len g ; :: thesis: (GoB g) * (i1,(width (GoB g))) in L~ g
then i + 1 <= len g by NAT_1:13;
then g /. i in LSeg (g,i) by ;
hence (GoB g) * (i1,(width (GoB g))) in L~ g by ; :: thesis: verum
end;
case i = len g ; :: thesis: (GoB g) * (i1,(width (GoB g))) in L~ g
then g /. i in LSeg (g,(i -' 1)) by ;
hence (GoB g) * (i1,(width (GoB g))) in L~ g by ; :: thesis: verum
end;
end;
end;
((GoB g) * (i1,(width (GoB g)))) `2 = N-bound (L~ g) by ;
then A158: ((GoB g) * (i1,(width (GoB g)))) `1 in { (q `1) where q is Point of () : ( q `2 = N-bound (L~ g) & q in L~ g ) } by A157;
for r being Real st r in B holds
r <= ((GoB g) * (i1,(width (GoB g)))) `1
proof
let r be Real; :: thesis: ( r in B implies r <= ((GoB g) * (i1,(width (GoB g)))) `1 )
assume r in B ; :: thesis: r <= ((GoB g) * (i1,(width (GoB g)))) `1
then ex q being Point of () st
( r = q `1 & q `2 = N-bound (L~ g) & q in L~ g ) ;
hence r <= ((GoB g) * (i1,(width (GoB g)))) `1 by Lm8; :: thesis: verum
end;
then A159: upper_bound B <= ((GoB g) * (i1,(width (GoB g)))) `1 by ;
((GoB g) * ((len (GoB g)),(width (GoB g)))) `1 is UpperBound of B
proof
let r be ExtReal; :: according to XXREAL_2:def 1 :: thesis: ( not r in B or r <= ((GoB g) * ((len (GoB g)),(width (GoB g)))) `1 )
assume r in B ; :: thesis: r <= ((GoB g) * ((len (GoB g)),(width (GoB g)))) `1
then A160: ex q being Point of () st
( r = q `1 & q `2 = N-bound (L~ g) & q in L~ g ) ;
1 <= width (GoB g) by GOBOARD7:33;
hence r <= ((GoB g) * ((len (GoB g)),(width (GoB g)))) `1 by ; :: thesis: verum
end;
then B is bounded_above ;
then upper_bound B >= ((GoB g) * (i1,(width (GoB g)))) `1 by ;
then ((GoB g) * (i1,(width (GoB g)))) `1 = upper_bound B by
.= upper_bound (proj1 | (N-most (L~ g))) by Th15 ;
hence ex b1 being Nat st
( [b1,(width (GoB g))] in Indices (GoB g) & (GoB g) * (b1,(width (GoB g))) = N-max (L~ g) ) by ; :: thesis: verum