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

set s1 = ((GoB g) * (1,1)) `2 ;

defpred S_{1}[ 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 : S_{1}[j] } ;

A1: { j where j is Element of NAT : S_{1}[j] } c= Seg (width (GoB g))
_{1}[j] } is Subset of NAT
from DOMAIN_1:sch 7();

A3: 1 <= len (GoB g) by GOBOARD7:32;

then consider i, j being 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 Th7;

j in NAT by ORDINAL1:def 12;

then j in { j where j is Element of NAT : S_{1}[j] }
by A4, A5, A6;

then reconsider Y = { j where j is Element of NAT : S_{1}[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 Nat st

( i in dom g & g /. i = (GoB g) * (1,j) ) ;

A10: min Y <= width (GoB g) by A7, A8, MATRIX_0:32;

A11: 1 <= len (GoB g) by A8, MATRIX_0:32;

1 <= min Y by A7, A8, MATRIX_0:32;

then A12: ((GoB g) * (1,(min Y))) `1 = ((GoB g) * (1,1)) `1 by A11, A10, GOBOARD5:2;

then A13: ((GoB g) * (1,(min Y))) `1 = W-bound (L~ g) by Th37;

consider i being 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_3:25;

A17: 1 <= i by A14, FINSEQ_3:25;

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 st r in B holds

r >= ((GoB g) * (1,(min Y))) `2

((GoB g) * (1,1)) `2 is LowerBound of B

then lower_bound B <= ((GoB g) * (1,(min Y))) `2 by A19, SEQ_4:def 2;

then ((GoB g) * (1,(min Y))) `2 = lower_bound B by A20, XXREAL_0:1

.= lower_bound (proj2 | (W-most (L~ g))) by Th13 ;

hence ex b_{1} being Nat st

( [1,b_{1}] in Indices (GoB g) & (GoB g) * (1,b_{1}) = W-min (L~ g) )
by A7, A8, A13, EUCLID:53; :: thesis: verum

proof

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 ;
let X be object ; :: 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 by XREAL_0:def 1; :: thesis: verum

end;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 by XREAL_0:def 1; :: thesis: verum

set s1 = ((GoB g) * (1,1)) `2 ;

defpred S

( i in dom g & g /. i = (GoB g) * (1,$1) ) );

set Y = { j where j is Element of NAT : S

A1: { j where j is Element of NAT : S

proof

A2:
{ j where j is Element of NAT : S
let y be object ; :: according to TARSKI:def 3 :: thesis: ( not y in { j where j is Element of NAT : S_{1}[j] } or y in Seg (width (GoB g)) )

assume y in { j where j is Element of NAT : S_{1}[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 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_0:def 4;

hence y in Seg (width (GoB g)) by ZFMISC_1:87; :: thesis: verum

end;assume y in { j where j is Element of NAT : S

then ex j being Element of NAT st

( y = j & [1,j] in Indices (GoB g) & ex i being 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_0:def 4;

hence y in Seg (width (GoB g)) by ZFMISC_1:87; :: thesis: verum

A3: 1 <= len (GoB g) by GOBOARD7:32;

then consider i, j being 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 Th7;

j in NAT by ORDINAL1:def 12;

then j in { j where j is Element of NAT : S

then reconsider Y = { j where j is Element of NAT : S

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 Nat st

( i in dom g & g /. i = (GoB g) * (1,j) ) ;

A10: min Y <= width (GoB g) by A7, A8, MATRIX_0:32;

A11: 1 <= len (GoB g) by A8, MATRIX_0:32;

1 <= min Y by A7, A8, MATRIX_0:32;

then A12: ((GoB g) * (1,(min Y))) `1 = ((GoB g) * (1,1)) `1 by A11, A10, GOBOARD5:2;

then A13: ((GoB g) * (1,(min Y))) `1 = W-bound (L~ g) by Th37;

consider i being 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_3:25;

A17: 1 <= i by A14, FINSEQ_3:25;

A18: now :: thesis: ( ( i < len g & (GoB g) * (1,(min Y)) in L~ g ) or ( i = len g & (GoB g) * (1,(min Y)) in L~ g ) )

((GoB g) * (1,(min Y))) `1 = W-bound (L~ g)
by A12, Th37;end;

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 st r in B holds

r >= ((GoB g) * (1,(min Y))) `2

proof

then A20:
lower_bound B >= ((GoB g) * (1,(min Y))) `2
by A19, SEQ_4:43;
let r be Real; :: 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;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

((GoB g) * (1,1)) `2 is LowerBound of B

proof

then
B is bounded_below
;
let r be ExtReal; :: 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, Th33; :: thesis: verum

end;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, Th33; :: thesis: verum

then lower_bound B <= ((GoB g) * (1,(min Y))) `2 by A19, SEQ_4:def 2;

then ((GoB g) * (1,(min Y))) `2 = lower_bound B by A20, XXREAL_0:1

.= lower_bound (proj2 | (W-most (L~ g))) by Th13 ;

hence ex b

( [1,b