let C be Simple_closed_curve; :: thesis: for n being Nat
for p being Point of (TOP-REAL 2)
for I being Integer st I = [\(((((p `1) - (W-bound C)) / ((E-bound C) - (W-bound C))) * (2 |^ n)) + 2)/] holds
p `1 < (W-bound C) + ((((E-bound C) - (W-bound C)) / (2 |^ n)) * (I - 1))
let n be Nat; :: thesis: for p being Point of (TOP-REAL 2)
for I being Integer st I = [\(((((p `1) - (W-bound C)) / ((E-bound C) - (W-bound C))) * (2 |^ n)) + 2)/] holds
p `1 < (W-bound C) + ((((E-bound C) - (W-bound C)) / (2 |^ n)) * (I - 1))
let p be Point of (TOP-REAL 2); :: thesis: for I being Integer st I = [\(((((p `1) - (W-bound C)) / ((E-bound C) - (W-bound C))) * (2 |^ n)) + 2)/] holds
p `1 < (W-bound C) + ((((E-bound C) - (W-bound C)) / (2 |^ n)) * (I - 1))
set W = W-bound C;
set E = E-bound C;
set EW = (E-bound C) - (W-bound C);
set PW = (p `1) - (W-bound C);
let I be Integer; :: thesis: ( I = [\(((((p `1) - (W-bound C)) / ((E-bound C) - (W-bound C))) * (2 |^ n)) + 2)/] implies p `1 < (W-bound C) + ((((E-bound C) - (W-bound C)) / (2 |^ n)) * (I - 1)) )
set KI = I - 1;
A1: 2 |^ n > 0 by NEWTON:83;
assume I = [\(((((p `1) - (W-bound C)) / ((E-bound C) - (W-bound C))) * (2 |^ n)) + 2)/] ; :: thesis: p `1 < (W-bound C) + ((((E-bound C) - (W-bound C)) / (2 |^ n)) * (I - 1))
then I > (((((p `1) - (W-bound C)) / ((E-bound C) - (W-bound C))) * (2 |^ n)) + 2) - 1 by INT_1:def 6;
then A2: I - 1 > (((((p `1) - (W-bound C)) / ((E-bound C) - (W-bound C))) * (2 |^ n)) + 1) - 1 by XREAL_1:9;
A3: (E-bound C) - (W-bound C) > 0 by TOPREAL5:17, XREAL_1:50;
then ((E-bound C) - (W-bound C)) / (2 |^ n) > 0 by A1, XREAL_1:139;
then A4: (((E-bound C) - (W-bound C)) / (2 |^ n)) * (I - 1) > (((E-bound C) - (W-bound C)) / (2 |^ n)) * ((((p `1) - (W-bound C)) / ((E-bound C) - (W-bound C))) * (2 |^ n)) by A2, XREAL_1:68;
A5: (W-bound C) + ((p `1) - (W-bound C)) = p `1 ;
(((E-bound C) - (W-bound C)) / (2 |^ n)) * ((((p `1) - (W-bound C)) / ((E-bound C) - (W-bound C))) * (2 |^ n)) = (p `1) - (W-bound C) by A3, A1, Lm2;
hence p `1 < (W-bound C) + ((((E-bound C) - (W-bound C)) / (2 |^ n)) * (I - 1)) by A5, A4, XREAL_1:6; :: thesis: verum