let X be RealUnitarySpace; :: thesis: for Y being Subset of X
for L being Functional of X holds
( Y is_summable_set_by L iff for e being Real st 0 < e holds
ex Y0 being finite Subset of X st
( not Y0 is empty & Y0 c= Y & ( for Y1 being finite Subset of X st not Y1 is empty & Y1 c= Y & Y0 misses Y1 holds
abs (setopfunc (Y1, the carrier of X,REAL,L,addreal)) < e ) ) )
let Y be Subset of X; :: thesis: for L being Functional of X holds
( Y is_summable_set_by L iff for e being Real st 0 < e holds
ex Y0 being finite Subset of X st
( not Y0 is empty & Y0 c= Y & ( for Y1 being finite Subset of X st not Y1 is empty & Y1 c= Y & Y0 misses Y1 holds
abs (setopfunc (Y1, the carrier of X,REAL,L,addreal)) < e ) ) )
let L be Functional of X; :: thesis: ( Y is_summable_set_by L iff for e being Real st 0 < e holds
ex Y0 being finite Subset of X st
( not Y0 is empty & Y0 c= Y & ( for Y1 being finite Subset of X st not Y1 is empty & Y1 c= Y & Y0 misses Y1 holds
abs (setopfunc (Y1, the carrier of X,REAL,L,addreal)) < e ) ) )
thus ( Y is_summable_set_by L implies for e being Real st 0 < e holds
ex Y0 being finite Subset of X st
( not Y0 is empty & Y0 c= Y & ( for Y1 being finite Subset of X st not Y1 is empty & Y1 c= Y & Y0 misses Y1 holds
abs (setopfunc (Y1, the carrier of X,REAL,L,addreal)) < e ) ) ) :: thesis: ( ( for e being Real st 0 < e holds
ex Y0 being finite Subset of X st
( not Y0 is empty & Y0 c= Y & ( for Y1 being finite Subset of X st not Y1 is empty & Y1 c= Y & Y0 misses Y1 holds
abs (setopfunc (Y1, the carrier of X,REAL,L,addreal)) < e ) ) ) implies Y is_summable_set_by L )
proof
assume Y is_summable_set_by L ; :: thesis: for e being Real st 0 < e holds
ex Y0 being finite Subset of X st
( not Y0 is empty & Y0 c= Y & ( for Y1 being finite Subset of X st not Y1 is empty & Y1 c= Y & Y0 misses Y1 holds
abs (setopfunc (Y1, the carrier of X,REAL,L,addreal)) < e ) )
then consider r being Real such that
A1: for e being Real st e > 0 holds
ex Y0 being finite Subset of X st
( not Y0 is empty & Y0 c= Y & ( for Y1 being finite Subset of X st Y0 c= Y1 & Y1 c= Y holds
abs (r - (setopfunc (Y1, the carrier of X,REAL,L,addreal))) < e ) ) by BHSP_6:def 6;
for e being Real st 0 < e holds
ex Y0 being finite Subset of X st
( not Y0 is empty & Y0 c= Y & ( for Y1 being finite Subset of X st not Y1 is empty & Y1 c= Y & Y0 misses Y1 holds
abs (setopfunc (Y1, the carrier of X,REAL,L,addreal)) < e ) )
proof
let e be Real; :: thesis: ( 0 < e implies ex Y0 being finite Subset of X st
( not Y0 is empty & Y0 c= Y & ( for Y1 being finite Subset of X st not Y1 is empty & Y1 c= Y & Y0 misses Y1 holds
abs (setopfunc (Y1, the carrier of X,REAL,L,addreal)) < e ) ) )
assume 0 < e ; :: thesis: ex Y0 being finite Subset of X st
( not Y0 is empty & Y0 c= Y & ( for Y1 being finite Subset of X st not Y1 is empty & Y1 c= Y & Y0 misses Y1 holds
abs (setopfunc (Y1, the carrier of X,REAL,L,addreal)) < e ) )
then consider Y0 being finite Subset of X such that
A2: not Y0 is empty and
A3: Y0 c= Y and
A4: for Y1 being finite Subset of X st Y0 c= Y1 & Y1 c= Y holds
abs (r - (setopfunc (Y1, the carrier of X,REAL,L,addreal))) < e / 2 by A1, XREAL_1:139;
for Y1 being finite Subset of X st not Y1 is empty & Y1 c= Y & Y0 misses Y1 holds
abs (setopfunc (Y1, the carrier of X,REAL,L,addreal)) < e
proof
let Y1 be finite Subset of X; :: thesis: ( not Y1 is empty & Y1 c= Y & Y0 misses Y1 implies abs (setopfunc (Y1, the carrier of X,REAL,L,addreal)) < e )
assume that
not Y1 is empty and
A5: Y1 c= Y and
A6: Y0 misses Y1 ; :: thesis: abs (setopfunc (Y1, the carrier of X,REAL,L,addreal)) < e
set Y19 = Y0 \/ Y1;
dom L = the carrier of X by FUNCT_2:def 1;
then setopfunc ((Y0 \/ Y1), the carrier of X,REAL,L,addreal) = addreal . ((setopfunc (Y0, the carrier of X,REAL,L,addreal)),(setopfunc (Y1, the carrier of X,REAL,L,addreal))) by A6, BHSP_5:14
.= (setopfunc (Y0, the carrier of X,REAL,L,addreal)) + (setopfunc (Y1, the carrier of X,REAL,L,addreal)) by BINOP_2:def 9 ;
then A7: setopfunc (Y1, the carrier of X,REAL,L,addreal) = (setopfunc ((Y0 \/ Y1), the carrier of X,REAL,L,addreal)) - (setopfunc (Y0, the carrier of X,REAL,L,addreal)) ;
Y0 c= Y0 \/ Y1 by XBOOLE_1:7;
then abs (r - (setopfunc ((Y0 \/ Y1), the carrier of X,REAL,L,addreal))) < e / 2 by A3, A4, A5, XBOOLE_1:8;
then A8: abs ((setopfunc ((Y0 \/ Y1), the carrier of X,REAL,L,addreal)) - r) < e / 2 by UNIFORM1:11;
abs (r - (setopfunc (Y0, the carrier of X,REAL,L,addreal))) < e / 2 by A3, A4;
hence abs (setopfunc (Y1, the carrier of X,REAL,L,addreal)) < e by A8, A7, Lm1; :: thesis: verum
end;
hence ex Y0 being finite Subset of X st
( not Y0 is empty & Y0 c= Y & ( for Y1 being finite Subset of X st not Y1 is empty & Y1 c= Y & Y0 misses Y1 holds
abs (setopfunc (Y1, the carrier of X,REAL,L,addreal)) < e ) ) by A2, A3; :: thesis: verum
end;
hence for e being Real st 0 < e holds
ex Y0 being finite Subset of X st
( not Y0 is empty & Y0 c= Y & ( for Y1 being finite Subset of X st not Y1 is empty & Y1 c= Y & Y0 misses Y1 holds
abs (setopfunc (Y1, the carrier of X,REAL,L,addreal)) < e ) ) ; :: thesis: verum
end;
assume A9: for e being Real st 0 < e holds
ex Y0 being finite Subset of X st
( not Y0 is empty & Y0 c= Y & ( for Y1 being finite Subset of X st not Y1 is empty & Y1 c= Y & Y0 misses Y1 holds
abs (setopfunc (Y1, the carrier of X,REAL,L,addreal)) < e ) ) ; :: thesis: Y is_summable_set_by L
ex r being Real st
for e being Real st 0 < e holds
ex Y0 being finite Subset of X st
( not Y0 is empty & Y0 c= Y & ( for Y1 being finite Subset of X st Y0 c= Y1 & Y1 c= Y holds
abs (r - (setopfunc (Y1, the carrier of X,REAL,L,addreal))) < e ) )
proof
defpred S1[ set , set ] means ( $2 is finite Subset of X & not $2 is empty & $2 c= Y & ( for z being Real st z = $1 holds
for Y1 being finite Subset of X st not Y1 is empty & Y1 c= Y & $2 misses Y1 holds
abs (setopfunc (Y1, the carrier of X,REAL,L,addreal)) < 1 / (z + 1) ) );
now
let x be set ; :: thesis: ( x in NAT implies ex y being set st
( y in bool Y & y is finite Subset of X & not y is empty & y c= Y & ( for z being Real st z = x holds
for Y1 being finite Subset of X st not Y1 is empty & Y1 c= Y & y misses Y1 holds
abs (setopfunc (Y1, the carrier of X,REAL,L,addreal)) < 1 / (z + 1) ) ) )
assume x in NAT ; :: thesis: ex y being set st
( y in bool Y & y is finite Subset of X & not y is empty & y c= Y & ( for z being Real st z = x holds
for Y1 being finite Subset of X st not Y1 is empty & Y1 c= Y & y misses Y1 holds
abs (setopfunc (Y1, the carrier of X,REAL,L,addreal)) < 1 / (z + 1) ) )
then reconsider xx = x as Element of NAT ;
reconsider e = 1 / (xx + 1) as Real ;
0 <= xx by NAT_1:2;
then 0 < xx + 1 by NAT_1:13;
then 0 / (xx + 1) < 1 / (xx + 1) by XREAL_1:74;
then consider Y0 being finite Subset of X such that
A10: not Y0 is empty and
A11: ( Y0 c= Y & ( for Y1 being finite Subset of X st not Y1 is empty & Y1 c= Y & Y0 misses Y1 holds
abs (setopfunc (Y1, the carrier of X,REAL,L,addreal)) < e ) ) by A9;
( Y0 in bool Y & ( for z being Real st z = x holds
for Y1 being finite Subset of X st not Y1 is empty & Y1 c= Y & Y0 misses Y1 holds
abs (setopfunc (Y1, the carrier of X,REAL,L,addreal)) < 1 / (z + 1) ) ) by A11, ZFMISC_1:def 1;
hence ex y being set st
( y in bool Y & y is finite Subset of X & not y is empty & y c= Y & ( for z being Real st z = x holds
for Y1 being finite Subset of X st not Y1 is empty & Y1 c= Y & y misses Y1 holds
abs (setopfunc (Y1, the carrier of X,REAL,L,addreal)) < 1 / (z + 1) ) ) by A10; :: thesis: verum
end;
then A12: for x being set st x in NAT holds
ex y being set st
( y in bool Y & S1[x,y] ) ;
A13: ex B being Function of NAT,(bool Y) st
for x being set st x in NAT holds
S1[x,B . x] from FUNCT_2:sch 1(A12);
 ex A being Function of NAT,(bool Y) st
for i being Element of NAT holds
( A . i is finite Subset of X & not A . i is empty & A . i c= Y & ( for Y1 being finite Subset of X st not Y1 is empty & Y1 c= Y & A . i misses Y1 holds
abs (setopfunc (Y1, the carrier of X,REAL,L,addreal)) < 1 / (i + 1) ) & ( for j being Element of NAT st i <= j holds
A . i c= A . j ) )
proof
consider B being Function of NAT,(bool Y) such that
A14: for x being set st x in NAT holds
( B . x is finite Subset of X & not B . x is empty & B . x c= Y & ( for z being Real st z = x holds
for Y1 being finite Subset of X st not Y1 is empty & Y1 c= Y & B . x misses Y1 holds
abs (setopfunc (Y1, the carrier of X,REAL,L,addreal)) < 1 / (z + 1) ) ) by A13;
deffunc H1( Nat, set ) -> set = (B . ($1 + 1)) \/ $2;
ex A being Function st
( dom A = NAT & A . 0 = B . 0 & ( for n being Nat holds A . (n + 1) = H1(n,A . n) ) ) from NAT_1:sch 11();
 then consider A being Function such that
A15: dom A = NAT and
A16: A . 0 = B . 0 and
A17: for n being Nat holds A . (n + 1) = (B . (n + 1)) \/ (A . n) ;
defpred S2[ Element of NAT ] means ( A . $1 is finite Subset of X & not A . $1 is empty & A . $1 c= Y & ( for Y1 being finite Subset of X st not Y1 is empty & Y1 c= Y & A . $1 misses Y1 holds
abs (setopfunc (Y1, the carrier of X,REAL,L,addreal)) < 1 / ($1 + 1) ) & ( for j being Element of NAT st $1 <= j holds
A . $1 c= A . j ) );
A18: now
let n be Element of NAT ; :: thesis: ( S2[n] implies S2[n + 1] )
assume A19: S2[n] ; :: thesis: S2[n + 1]
A20: for Y1 being finite Subset of X st not Y1 is empty & Y1 c= Y & A . (n + 1) misses Y1 holds
abs (setopfunc (Y1, the carrier of X,REAL,L,addreal)) < 1 / ((n + 1) + 1)
proof
let Y1 be finite Subset of X; :: thesis: ( not Y1 is empty & Y1 c= Y & A . (n + 1) misses Y1 implies abs (setopfunc (Y1, the carrier of X,REAL,L,addreal)) < 1 / ((n + 1) + 1) )
assume that
A21: ( not Y1 is empty & Y1 c= Y ) and
A22: A . (n + 1) misses Y1 ; :: thesis: abs (setopfunc (Y1, the carrier of X,REAL,L,addreal)) < 1 / ((n + 1) + 1)
A . (n + 1) = (B . (n + 1)) \/ (A . n) by A17;
then B . (n + 1) misses Y1 by A22, XBOOLE_1:7, XBOOLE_1:63;
hence abs (setopfunc (Y1, the carrier of X,REAL,L,addreal)) < 1 / ((n + 1) + 1) by A14, A21; :: thesis: verum
end;
defpred S3[ Element of NAT ] means ( n + 1 <= $1 implies A . (n + 1) c= A . $1 );
A23: for j being Element of NAT st S3[j] holds
S3[j + 1]
proof
let j be Element of NAT ; :: thesis: ( S3[j] implies S3[j + 1] )
assume that
A24: S3[j] and
A25: n + 1 <= j + 1 ; :: thesis: A . (n + 1) c= A . (j + 1)
now
per cases ( n = j or n <> j ) ;
case n = j ; :: thesis: A . (n + 1) c= A . (j + 1)
hence A . (n + 1) c= A . (j + 1) ; :: thesis: verum
end;
case A26: n <> j ; :: thesis: A . (n + 1) c= A . (j + 1)
A . (j + 1) = (B . (j + 1)) \/ (A . j) by A17;
then A27: A . j c= A . (j + 1) by XBOOLE_1:7;
n <= j by A25, XREAL_1:6;
then n < j by A26, XXREAL_0:1;
hence A . (n + 1) c= A . (j + 1) by A24, A27, NAT_1:13, XBOOLE_1:1; :: thesis: verum
end;
end;
end;
hence A . (n + 1) c= A . (j + 1) ; :: thesis: verum
end;
A28: S3[ 0 ] by NAT_1:3;
A29: for j being Element of NAT holds S3[j] from NAT_1:sch 1(A28, A23);
 ( A . (n + 1) = (B . (n + 1)) \/ (A . n) & B . (n + 1) is finite Subset of X ) by A14, A17;
hence S2[n + 1] by A19, A20, A29, XBOOLE_1:8; :: thesis: verum
end;
for j0 being Element of NAT st j0 = 0 holds
for j being Element of NAT st j0 <= j holds
A . j0 c= A . j
proof
let j0 be Element of NAT ; :: thesis: ( j0 = 0 implies for j being Element of NAT st j0 <= j holds
A . j0 c= A . j )
assume A30: j0 = 0 ; :: thesis: for j being Element of NAT st j0 <= j holds
A . j0 c= A . j
defpred S3[ Element of NAT ] means ( j0 <= $1 implies A . j0 c= A . $1 );
A31: now
let j be Element of NAT ; :: thesis: ( S3[j] implies S3[j + 1] )
assume A32: S3[j] ; :: thesis: S3[j + 1]
A . (j + 1) = (B . (j + 1)) \/ (A . j) by A17;
then A . j c= A . (j + 1) by XBOOLE_1:7;
hence S3[j + 1] by A30, A32, NAT_1:2, XBOOLE_1:1; :: thesis: verum
end;
A33: S3[ 0 ] by A30;
thus for j being Element of NAT holds S3[j] from NAT_1:sch 1(A33, A31); :: thesis: verum
end;
then A34: S2[ 0 ] by A14, A16;
A35: for i being Element of NAT holds S2[i] from NAT_1:sch 1(A34, A18);
 rng A c= bool Y
proof
let y be set ; :: according to TARSKI:def 3 :: thesis: ( not y in rng A or y in bool Y )
assume y in rng A ; :: thesis: y in bool Y
then consider x being set such that
A36: x in dom A and
A37: y = A . x by FUNCT_1:def 3;
reconsider i = x as Element of NAT by A15, A36;
A . i c= Y by A35;
hence y in bool Y by A37, ZFMISC_1:def 1; :: thesis: verum
end;
then A is Function of NAT,(bool Y) by A15, FUNCT_2:2;
hence ex A being Function of NAT,(bool Y) st
for i being Element of NAT holds
( A . i is finite Subset of X & not A . i is empty & A . i c= Y & ( for Y1 being finite Subset of X st not Y1 is empty & Y1 c= Y & A . i misses Y1 holds
abs (setopfunc (Y1, the carrier of X,REAL,L,addreal)) < 1 / (i + 1) ) & ( for j being Element of NAT st i <= j holds
A . i c= A . j ) ) by A35; :: thesis: verum
end;
then consider A being Function of NAT,(bool Y) such that
A38: for i being Element of NAT holds
( A . i is finite Subset of X & not A . i is empty & A . i c= Y & ( for Y1 being finite Subset of X st not Y1 is empty & Y1 c= Y & A . i misses Y1 holds
abs (setopfunc (Y1, the carrier of X,REAL,L,addreal)) < 1 / (i + 1) ) & ( for j being Element of NAT st i <= j holds
A . i c= A . j ) ) ;
defpred S2[ set , set ] means ex Y1 being finite Subset of X st
( not Y1 is empty & A . $1 = Y1 & $2 = setopfunc (Y1, the carrier of X,REAL,L,addreal) );
A39: for x being set st x in NAT holds
ex y being set st
( y in REAL & S2[x,y] )
proof
let x be set ; :: thesis: ( x in NAT implies ex y being set st
( y in REAL & S2[x,y] ) )
assume x in NAT ; :: thesis: ex y being set st
( y in REAL & S2[x,y] )
then reconsider i = x as Element of NAT ;
A . i is finite Subset of X by A38;
then reconsider Y1 = A . x as finite Subset of X ;
reconsider y = setopfunc (Y1, the carrier of X,REAL,L,addreal) as set ;
not A . i is empty by A38;
then ex Y1 being finite Subset of X st
( not Y1 is empty & A . x = Y1 & y = setopfunc (Y1, the carrier of X,REAL,L,addreal) ) ;
hence ex y being set st
( y in REAL & S2[x,y] ) ; :: thesis: verum
end;
ex F being Function of NAT,REAL st
for x being set st x in NAT holds
S2[x,F . x] from FUNCT_2:sch 1(A39);
 then consider F being Function of NAT,REAL such that
A40: for x being set st x in NAT holds
ex Y1 being finite Subset of X st
( not Y1 is empty & A . x = Y1 & F . x = setopfunc (Y1, the carrier of X,REAL,L,addreal) ) ;
set seq = F;
A41: for e being real number st e > 0 holds
ex k being Element of NAT st
for n, m being Element of NAT st n >= k & m >= k holds
abs ((F . n) - (F . m)) < e
proof
let e be real number ; :: thesis: ( e > 0 implies ex k being Element of NAT st
for n, m being Element of NAT st n >= k & m >= k holds
abs ((F . n) - (F . m)) < e )
assume A42: e > 0 ; :: thesis: ex k being Element of NAT st
for n, m being Element of NAT st n >= k & m >= k holds
abs ((F . n) - (F . m)) < e
A43: e / 2 > 0 / 2 by A42, XREAL_1:74;
then consider k being Element of NAT such that
A44: 1 / (k + 1) < e / 2 by Lm2;
take k ; :: thesis: for n, m being Element of NAT st n >= k & m >= k holds
abs ((F . n) - (F . m)) < e
let n, m be Element of NAT ; :: thesis: ( n >= k & m >= k implies abs ((F . n) - (F . m)) < e )
assume that
A45: n >= k and
A46: m >= k ; :: thesis: abs ((F . n) - (F . m)) < e
consider Y2 being finite Subset of X such that
not Y2 is empty and
A47: A . m = Y2 and
A48: F . m = setopfunc (Y2, the carrier of X,REAL,L,addreal) by A40;
consider Y0 being finite Subset of X such that
not Y0 is empty and
A49: A . k = Y0 and
F . k = setopfunc (Y0, the carrier of X,REAL,L,addreal) by A40;
A50: Y0 c= Y2 by A38, A46, A49, A47;
consider Y1 being finite Subset of X such that
not Y1 is empty and
A51: A . n = Y1 and
A52: F . n = setopfunc (Y1, the carrier of X,REAL,L,addreal) by A40;
A53: Y0 c= Y1 by A38, A45, A49, A51;
now
per cases ( Y0 = Y1 or Y0 <> Y1 ) ;
case A54: Y0 = Y1 ; :: thesis: abs ((F . n) - (F . m)) < e
now
per cases ( Y0 = Y2 or Y0 <> Y2 ) ;
case Y0 = Y2 ; :: thesis: abs ((F . n) - (F . m)) < e
hence abs ((F . n) - (F . m)) < e by A42, A52, A48, A54, ABSVALUE:2; :: thesis: verum
end;
case A55: Y0 <> Y2 ; :: thesis: abs ((F . n) - (F . m)) < e
ex Y02 being finite Subset of X st
( not Y02 is empty & Y02 c= Y & Y02 misses Y0 & Y0 \/ Y02 = Y2 )
proof
take Y02 = Y2 \ Y0; :: thesis: ( not Y02 is empty & Y02 c= Y & Y02 misses Y0 & Y0 \/ Y02 = Y2 )
A56: Y2 \ Y0 c= Y2 by XBOOLE_1:36;
Y0 \/ Y02 = Y0 \/ Y2 by XBOOLE_1:39
.= Y2 by A50, XBOOLE_1:12 ;
hence ( not Y02 is empty & Y02 c= Y & Y02 misses Y0 & Y0 \/ Y02 = Y2 ) by A47, A55, A56, XBOOLE_1:1, XBOOLE_1:79; :: thesis: verum
end;
then consider Y02 being finite Subset of X such that
A57: ( not Y02 is empty & Y02 c= Y ) and
A58: Y02 misses Y0 and
A59: Y0 \/ Y02 = Y2 ;
abs (setopfunc (Y02, the carrier of X,REAL,L,addreal)) < 1 / (k + 1) by A38, A49, A57, A58;
then A60: abs (setopfunc (Y02, the carrier of X,REAL,L,addreal)) < e / 2 by A44, XXREAL_0:2;
dom L = the carrier of X by FUNCT_2:def 1;
then setopfunc (Y2, the carrier of X,REAL,L,addreal) = addreal . ((setopfunc (Y0, the carrier of X,REAL,L,addreal)),(setopfunc (Y02, the carrier of X,REAL,L,addreal))) by A58, A59, BHSP_5:14
.= (setopfunc (Y0, the carrier of X,REAL,L,addreal)) + (setopfunc (Y02, the carrier of X,REAL,L,addreal)) by BINOP_2:def 9 ;
then A61: abs ((F . n) - (F . m)) = abs (- (setopfunc (Y02, the carrier of X,REAL,L,addreal))) by A52, A48, A54
.= abs (setopfunc (Y02, the carrier of X,REAL,L,addreal)) by COMPLEX1:52 ;
e / 2 < e by A42, XREAL_1:216;
hence abs ((F . n) - (F . m)) < e by A60, A61, XXREAL_0:2; :: thesis: verum
end;
end;
end;
hence abs ((F . n) - (F . m)) < e ; :: thesis: verum
end;
case A62: Y0 <> Y1 ; :: thesis: abs ((F . n) - (F . m)) < e
now
per cases ( Y0 = Y2 or Y0 <> Y2 ) ;
case A63: Y0 = Y2 ; :: thesis: abs ((F . n) - (F . m)) < e
ex Y01 being finite Subset of X st
( not Y01 is empty & Y01 c= Y & Y01 misses Y0 & Y0 \/ Y01 = Y1 )
proof
take Y01 = Y1 \ Y0; :: thesis: ( not Y01 is empty & Y01 c= Y & Y01 misses Y0 & Y0 \/ Y01 = Y1 )
A64: Y1 \ Y0 c= Y1 by XBOOLE_1:36;
Y0 \/ Y01 = Y0 \/ Y1 by XBOOLE_1:39
.= Y1 by A53, XBOOLE_1:12 ;
hence ( not Y01 is empty & Y01 c= Y & Y01 misses Y0 & Y0 \/ Y01 = Y1 ) by A51, A62, A64, XBOOLE_1:1, XBOOLE_1:79; :: thesis: verum
end;
then consider Y01 being finite Subset of X such that
A65: ( not Y01 is empty & Y01 c= Y ) and
A66: Y01 misses Y0 and
A67: Y0 \/ Y01 = Y1 ;
dom L = the carrier of X by FUNCT_2:def 1;
then A68: setopfunc (Y1, the carrier of X,REAL,L,addreal) = addreal . ((setopfunc (Y0, the carrier of X,REAL,L,addreal)),(setopfunc (Y01, the carrier of X,REAL,L,addreal))) by A66, A67, BHSP_5:14
.= (setopfunc (Y0, the carrier of X,REAL,L,addreal)) + (setopfunc (Y01, the carrier of X,REAL,L,addreal)) by BINOP_2:def 9 ;
abs (setopfunc (Y01, the carrier of X,REAL,L,addreal)) < 1 / (k + 1) by A38, A49, A65, A66;
then abs ((F . n) - (F . m)) < e / 2 by A44, A52, A48, A63, A68, XXREAL_0:2;
then (abs ((F . n) - (F . m))) + 0 < (e / 2) + (e / 2) by A43, XREAL_1:8;
hence abs ((F . n) - (F . m)) < e ; :: thesis: verum
end;
case A69: Y0 <> Y2 ; :: thesis: abs ((F . n) - (F . m)) < e
ex Y02 being finite Subset of X st
( not Y02 is empty & Y02 c= Y & Y02 misses Y0 & Y0 \/ Y02 = Y2 )
proof
take Y02 = Y2 \ Y0; :: thesis: ( not Y02 is empty & Y02 c= Y & Y02 misses Y0 & Y0 \/ Y02 = Y2 )
A70: Y2 \ Y0 c= Y2 by XBOOLE_1:36;
Y0 \/ Y02 = Y0 \/ Y2 by XBOOLE_1:39
.= Y2 by A50, XBOOLE_1:12 ;
hence ( not Y02 is empty & Y02 c= Y & Y02 misses Y0 & Y0 \/ Y02 = Y2 ) by A47, A69, A70, XBOOLE_1:1, XBOOLE_1:79; :: thesis: verum
end;
then consider Y02 being finite Subset of X such that
A71: ( not Y02 is empty & Y02 c= Y ) and
A72: Y02 misses Y0 and
A73: Y0 \/ Y02 = Y2 ;
dom L = the carrier of X by FUNCT_2:def 1;
then A74: setopfunc (Y2, the carrier of X,REAL,L,addreal) = addreal . ((setopfunc (Y0, the carrier of X,REAL,L,addreal)),(setopfunc (Y02, the carrier of X,REAL,L,addreal))) by A72, A73, BHSP_5:14
.= (setopfunc (Y0, the carrier of X,REAL,L,addreal)) + (setopfunc (Y02, the carrier of X,REAL,L,addreal)) by BINOP_2:def 9 ;
ex Y01 being finite Subset of X st
( not Y01 is empty & Y01 c= Y & Y01 misses Y0 & Y0 \/ Y01 = Y1 )
proof
take Y01 = Y1 \ Y0; :: thesis: ( not Y01 is empty & Y01 c= Y & Y01 misses Y0 & Y0 \/ Y01 = Y1 )
A75: Y1 \ Y0 c= Y1 by XBOOLE_1:36;
Y0 \/ Y01 = Y0 \/ Y1 by XBOOLE_1:39
.= Y1 by A53, XBOOLE_1:12 ;
hence ( not Y01 is empty & Y01 c= Y & Y01 misses Y0 & Y0 \/ Y01 = Y1 ) by A51, A62, A75, XBOOLE_1:1, XBOOLE_1:79; :: thesis: verum
end;
then consider Y01 being finite Subset of X such that
A76: ( not Y01 is empty & Y01 c= Y ) and
A77: Y01 misses Y0 and
A78: Y0 \/ Y01 = Y1 ;
dom L = the carrier of X by FUNCT_2:def 1;
then setopfunc (Y1, the carrier of X,REAL,L,addreal) = addreal . ((setopfunc (Y0, the carrier of X,REAL,L,addreal)),(setopfunc (Y01, the carrier of X,REAL,L,addreal))) by A77, A78, BHSP_5:14
.= (setopfunc (Y0, the carrier of X,REAL,L,addreal)) + (setopfunc (Y01, the carrier of X,REAL,L,addreal)) by BINOP_2:def 9 ;
then (F . n) - (F . m) = (setopfunc (Y01, the carrier of X,REAL,L,addreal)) - (setopfunc (Y02, the carrier of X,REAL,L,addreal)) by A52, A48, A74;
then A79: abs ((F . n) - (F . m)) <= (abs (setopfunc (Y01, the carrier of X,REAL,L,addreal))) + (abs (setopfunc (Y02, the carrier of X,REAL,L,addreal))) by COMPLEX1:57;
abs (setopfunc (Y02, the carrier of X,REAL,L,addreal)) < 1 / (k + 1) by A38, A49, A71, A72;
then A80: abs (setopfunc (Y02, the carrier of X,REAL,L,addreal)) < e / 2 by A44, XXREAL_0:2;
abs (setopfunc (Y01, the carrier of X,REAL,L,addreal)) < 1 / (k + 1) by A38, A49, A76, A77;
then abs (setopfunc (Y01, the carrier of X,REAL,L,addreal)) < e / 2 by A44, XXREAL_0:2;
then (abs (setopfunc (Y01, the carrier of X,REAL,L,addreal))) + (abs (setopfunc (Y02, the carrier of X,REAL,L,addreal))) < (e / 2) + (e / 2) by A80, XREAL_1:8;
hence abs ((F . n) - (F . m)) < e by A79, XXREAL_0:2; :: thesis: verum
end;
end;
end;
hence abs ((F . n) - (F . m)) < e ; :: thesis: verum
end;
end;
end;
hence abs ((F . n) - (F . m)) < e ; :: thesis: verum
end;
for e being real number st 0 < e holds
ex k being Element of NAT st
for m being Element of NAT st k <= m holds
abs ((F . m) - (F . k)) < e
proof
let e be real number ; :: thesis: ( 0 < e implies ex k being Element of NAT st
for m being Element of NAT st k <= m holds
abs ((F . m) - (F . k)) < e )
assume 0 < e ; :: thesis: ex k being Element of NAT st
for m being Element of NAT st k <= m holds
abs ((F . m) - (F . k)) < e
then consider k being Element of NAT such that
A81: for n, m being Element of NAT st n >= k & m >= k holds
abs ((F . n) - (F . m)) < e by A41;
for m being Element of NAT st k <= m holds
abs ((F . m) - (F . k)) < e by A81;
hence ex k being Element of NAT st
for m being Element of NAT st k <= m holds
abs ((F . m) - (F . k)) < e ; :: thesis: verum
end;
then F is convergent by SEQ_4:41;
then consider x being real number such that
A82: for r being real number st r > 0 holds
ex m being Element of NAT st
for n being Element of NAT st n >= m holds
abs ((F . n) - x) < r by SEQ_2:def 6;
reconsider r = x as Real by XREAL_0:def 1;
take r ; :: thesis: for e being Real st 0 < e holds
ex Y0 being finite Subset of X st
( not Y0 is empty & Y0 c= Y & ( for Y1 being finite Subset of X st Y0 c= Y1 & Y1 c= Y holds
abs (r - (setopfunc (Y1, the carrier of X,REAL,L,addreal))) < e ) )
for e being Real st 0 < e holds
ex Y0 being finite Subset of X st
( not Y0 is empty & Y0 c= Y & ( for Y1 being finite Subset of X st Y0 c= Y1 & Y1 c= Y holds
abs (r - (setopfunc (Y1, the carrier of X,REAL,L,addreal))) < e ) )
proof
let e be Real; :: thesis: ( 0 < e implies ex Y0 being finite Subset of X st
( not Y0 is empty & Y0 c= Y & ( for Y1 being finite Subset of X st Y0 c= Y1 & Y1 c= Y holds
abs (r - (setopfunc (Y1, the carrier of X,REAL,L,addreal))) < e ) ) )
assume e > 0 ; :: thesis: ex Y0 being finite Subset of X st
( not Y0 is empty & Y0 c= Y & ( for Y1 being finite Subset of X st Y0 c= Y1 & Y1 c= Y holds
abs (r - (setopfunc (Y1, the carrier of X,REAL,L,addreal))) < e ) )
then A83: e / 2 > 0 / 2 by XREAL_1:74;
then consider m being Element of NAT such that
A84: for n being Element of NAT st n >= m holds
abs ((F . n) - r) < e / 2 by A82;
consider i being Element of NAT such that
A85: 1 / (i + 1) < e / 2 and
A86: i >= m by A83, Lm3;
consider Y0 being finite Subset of X such that
A87: not Y0 is empty and
A88: A . i = Y0 and
A89: F . i = setopfunc (Y0, the carrier of X,REAL,L,addreal) by A40;
A90: abs ((setopfunc (Y0, the carrier of X,REAL,L,addreal)) - r) < e / 2 by A84, A86, A89;
now
let Y1 be finite Subset of X; :: thesis: ( Y0 c= Y1 & Y1 c= Y implies abs (r - (setopfunc (Y1, the carrier of X,REAL,L,addreal))) < e )
assume that
A91: Y0 c= Y1 and
A92: Y1 c= Y ; :: thesis: abs (r - (setopfunc (Y1, the carrier of X,REAL,L,addreal))) < e
now
per cases ( Y0 = Y1 or Y0 <> Y1 ) ;
case Y0 = Y1 ; :: thesis: abs (r - (setopfunc (Y1, the carrier of X,REAL,L,addreal))) < e
then abs (r - (setopfunc (Y1, the carrier of X,REAL,L,addreal))) < e / 2 by A90, UNIFORM1:11;
then (abs (x - (setopfunc (Y1, the carrier of X,REAL,L,addreal)))) + 0 < (e / 2) + (e / 2) by A83, XREAL_1:8;
hence abs (r - (setopfunc (Y1, the carrier of X,REAL,L,addreal))) < e ; :: thesis: verum
end;
case A93: Y0 <> Y1 ; :: thesis: abs (r - (setopfunc (Y1, the carrier of X,REAL,L,addreal))) < e
ex Y2 being finite Subset of X st
( not Y2 is empty & Y2 c= Y & Y0 misses Y2 & Y0 \/ Y2 = Y1 )
proof
take Y2 = Y1 \ Y0; :: thesis: ( not Y2 is empty & Y2 c= Y & Y0 misses Y2 & Y0 \/ Y2 = Y1 )
A94: Y1 \ Y0 c= Y1 by XBOOLE_1:36;
Y0 \/ Y2 = Y0 \/ Y1 by XBOOLE_1:39
.= Y1 by A91, XBOOLE_1:12 ;
hence ( not Y2 is empty & Y2 c= Y & Y0 misses Y2 & Y0 \/ Y2 = Y1 ) by A92, A93, A94, XBOOLE_1:1, XBOOLE_1:79; :: thesis: verum
end;
then consider Y2 being finite Subset of X such that
A95: ( not Y2 is empty & Y2 c= Y ) and
A96: Y0 misses Y2 and
A97: Y0 \/ Y2 = Y1 ;
dom L = the carrier of X by FUNCT_2:def 1;
then (setopfunc (Y1, the carrier of X,REAL,L,addreal)) - r = (addreal . ((setopfunc (Y0, the carrier of X,REAL,L,addreal)),(setopfunc (Y2, the carrier of X,REAL,L,addreal)))) - r by A96, A97, BHSP_5:14
.= ((setopfunc (Y0, the carrier of X,REAL,L,addreal)) + (setopfunc (Y2, the carrier of X,REAL,L,addreal))) - r by BINOP_2:def 9
.= ((setopfunc (Y0, the carrier of X,REAL,L,addreal)) - r) + (setopfunc (Y2, the carrier of X,REAL,L,addreal)) ;
then abs ((setopfunc (Y1, the carrier of X,REAL,L,addreal)) - r) <= (abs ((setopfunc (Y0, the carrier of X,REAL,L,addreal)) - r)) + (abs (setopfunc (Y2, the carrier of X,REAL,L,addreal))) by ABSVALUE:9;
then A98: abs (x - (setopfunc (Y1, the carrier of X,REAL,L,addreal))) <= (abs ((setopfunc (Y0, the carrier of X,REAL,L,addreal)) - r)) + (abs (setopfunc (Y2, the carrier of X,REAL,L,addreal))) by UNIFORM1:11;
abs (setopfunc (Y2, the carrier of X,REAL,L,addreal)) < 1 / (i + 1) by A38, A88, A95, A96;
then abs (setopfunc (Y2, the carrier of X,REAL,L,addreal)) < e / 2 by A85, XXREAL_0:2;
then (abs ((setopfunc (Y0, the carrier of X,REAL,L,addreal)) - r)) + (abs (setopfunc (Y2, the carrier of X,REAL,L,addreal))) < (e / 2) + (e / 2) by A90, XREAL_1:8;
hence abs (r - (setopfunc (Y1, the carrier of X,REAL,L,addreal))) < e by A98, XXREAL_0:2; :: thesis: verum
end;
end;
end;
hence abs (r - (setopfunc (Y1, the carrier of X,REAL,L,addreal))) < e ; :: thesis: verum
end;
hence ex Y0 being finite Subset of X st
( not Y0 is empty & Y0 c= Y & ( for Y1 being finite Subset of X st Y0 c= Y1 & Y1 c= Y holds
abs (r - (setopfunc (Y1, the carrier of X,REAL,L,addreal))) < e ) ) by A87, A88; :: thesis: verum
end;
hence for e being Real st 0 < e holds
ex Y0 being finite Subset of X st
( not Y0 is empty & Y0 c= Y & ( for Y1 being finite Subset of X st Y0 c= Y1 & Y1 c= Y holds
abs (r - (setopfunc (Y1, the carrier of X,REAL,L,addreal))) < e ) ) ; :: thesis: verum
end;
hence Y is_summable_set_by L by BHSP_6:def 6; :: thesis: verum