let I be non empty closed_interval Subset of REAL; :: thesis: for TD being tagged_division of I
for jauge being positive-yielding Function of I,REAL
for r1, r2, s being Real
for D1 being Division of I
for fc, f being Function of I,REAL
for epsilon being Real st fc = chi (I,I) & r1 = min (rng (upper_volume (fc,D1))) & r2 = epsilon / ((2 * (len D1)) * |.((upper_bound (rng f)) - (lower_bound (rng f))).|) & 0 < r1 & 0 < r2 & s = (min (r1,r2)) / 2 & jauge = s (#) fc & TD is jauge -fine holds
( delta (division_of TD) < min (rng (upper_volume (fc,D1))) & delta (division_of TD) < epsilon / ((2 * (len D1)) * |.((upper_bound (rng f)) - (lower_bound (rng f))).|) )
let TD be tagged_division of I; :: thesis: for jauge being positive-yielding Function of I,REAL
for r1, r2, s being Real
for D1 being Division of I
for fc, f being Function of I,REAL
for epsilon being Real st fc = chi (I,I) & r1 = min (rng (upper_volume (fc,D1))) & r2 = epsilon / ((2 * (len D1)) * |.((upper_bound (rng f)) - (lower_bound (rng f))).|) & 0 < r1 & 0 < r2 & s = (min (r1,r2)) / 2 & jauge = s (#) fc & TD is jauge -fine holds
( delta (division_of TD) < min (rng (upper_volume (fc,D1))) & delta (division_of TD) < epsilon / ((2 * (len D1)) * |.((upper_bound (rng f)) - (lower_bound (rng f))).|) )
let jauge be positive-yielding Function of I,REAL; :: thesis: for r1, r2, s being Real
for D1 being Division of I
for fc, f being Function of I,REAL
for epsilon being Real st fc = chi (I,I) & r1 = min (rng (upper_volume (fc,D1))) & r2 = epsilon / ((2 * (len D1)) * |.((upper_bound (rng f)) - (lower_bound (rng f))).|) & 0 < r1 & 0 < r2 & s = (min (r1,r2)) / 2 & jauge = s (#) fc & TD is jauge -fine holds
( delta (division_of TD) < min (rng (upper_volume (fc,D1))) & delta (division_of TD) < epsilon / ((2 * (len D1)) * |.((upper_bound (rng f)) - (lower_bound (rng f))).|) )
let r1, r2, s be Real; :: thesis: for D1 being Division of I
for fc, f being Function of I,REAL
for epsilon being Real st fc = chi (I,I) & r1 = min (rng (upper_volume (fc,D1))) & r2 = epsilon / ((2 * (len D1)) * |.((upper_bound (rng f)) - (lower_bound (rng f))).|) & 0 < r1 & 0 < r2 & s = (min (r1,r2)) / 2 & jauge = s (#) fc & TD is jauge -fine holds
( delta (division_of TD) < min (rng (upper_volume (fc,D1))) & delta (division_of TD) < epsilon / ((2 * (len D1)) * |.((upper_bound (rng f)) - (lower_bound (rng f))).|) )
let D1 be Division of I; :: thesis: for fc, f being Function of I,REAL
for epsilon being Real st fc = chi (I,I) & r1 = min (rng (upper_volume (fc,D1))) & r2 = epsilon / ((2 * (len D1)) * |.((upper_bound (rng f)) - (lower_bound (rng f))).|) & 0 < r1 & 0 < r2 & s = (min (r1,r2)) / 2 & jauge = s (#) fc & TD is jauge -fine holds
( delta (division_of TD) < min (rng (upper_volume (fc,D1))) & delta (division_of TD) < epsilon / ((2 * (len D1)) * |.((upper_bound (rng f)) - (lower_bound (rng f))).|) )
let fc, f be Function of I,REAL; :: thesis: for epsilon being Real st fc = chi (I,I) & r1 = min (rng (upper_volume (fc,D1))) & r2 = epsilon / ((2 * (len D1)) * |.((upper_bound (rng f)) - (lower_bound (rng f))).|) & 0 < r1 & 0 < r2 & s = (min (r1,r2)) / 2 & jauge = s (#) fc & TD is jauge -fine holds
( delta (division_of TD) < min (rng (upper_volume (fc,D1))) & delta (division_of TD) < epsilon / ((2 * (len D1)) * |.((upper_bound (rng f)) - (lower_bound (rng f))).|) )
let epsilon be Real; :: thesis: ( fc = chi (I,I) & r1 = min (rng (upper_volume (fc,D1))) & r2 = epsilon / ((2 * (len D1)) * |.((upper_bound (rng f)) - (lower_bound (rng f))).|) & 0 < r1 & 0 < r2 & s = (min (r1,r2)) / 2 & jauge = s (#) fc & TD is jauge -fine implies ( delta (division_of TD) < min (rng (upper_volume (fc,D1))) & delta (division_of TD) < epsilon / ((2 * (len D1)) * |.((upper_bound (rng f)) - (lower_bound (rng f))).|) ) )
assume that
A1: fc = chi (I,I) and
A2: r1 = min (rng (upper_volume (fc,D1))) and
A3: r2 = epsilon / ((2 * (len D1)) * |.((upper_bound (rng f)) - (lower_bound (rng f))).|) and
A4: 0 < r1 and
A5: 0 < r2 and
A6: s = (min (r1,r2)) / 2 and
A7: jauge = s (#) fc and
A8: TD is jauge -fine ; :: thesis: ( delta (division_of TD) < min (rng (upper_volume (fc,D1))) & delta (division_of TD) < epsilon / ((2 * (len D1)) * |.((upper_bound (rng f)) - (lower_bound (rng f))).|) )
A9: delta (division_of TD) <= (min (r1,r2)) / 2 by A1, A6, A7, A8, Th42;
(min (r1,r2)) / 2 < (min (r1,r2)) / 1 by A4, A5, XREAL_1:76;
then delta (division_of TD) < min (r1,r2) by A9, XXREAL_0:2;
hence ( delta (division_of TD) < min (rng (upper_volume (fc,D1))) & delta (division_of TD) < epsilon / ((2 * (len D1)) * |.((upper_bound (rng f)) - (lower_bound (rng f))).|) ) by A2, A3, XXREAL_0:23; :: thesis: verum