let C be Simple_closed_curve; :: thesis: for n being Element of 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
(W-bound C) + ((((E-bound C) - (W-bound C)) / (2 |^ n)) * (I - 2)) <= p `1
let n be Element of 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
(W-bound C) + ((((E-bound C) - (W-bound C)) / (2 |^ n)) * (I - 2)) <= p `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
(W-bound C) + ((((E-bound C) - (W-bound C)) / (2 |^ n)) * (I - 2)) <= p `1
set E = E-bound C;
set W = W-bound C;
set EW = (E-bound C) - (W-bound C);
set PW = (p `1 ) - (W-bound C);
set KI = [\((((p `1 ) - (W-bound C)) / ((E-bound C) - (W-bound C))) * (2 |^ n))/];
let I be Integer; :: thesis: ( I = [\(((((p `1 ) - (W-bound C)) / ((E-bound C) - (W-bound C))) * (2 |^ n)) + 2)/] implies (W-bound C) + ((((E-bound C) - (W-bound C)) / (2 |^ n)) * (I - 2)) <= p `1 )
assume I = [\(((((p `1 ) - (W-bound C)) / ((E-bound C) - (W-bound C))) * (2 |^ n)) + 2)/] ; :: thesis: (W-bound C) + ((((E-bound C) - (W-bound C)) / (2 |^ n)) * (I - 2)) <= p `1
then A1: I - 2 = [\((((p `1 ) - (W-bound C)) / ((E-bound C) - (W-bound C))) * (2 |^ n))/] by Lm1;
E-bound C > W-bound C by TOPREAL5:23;
then A2: ( (E-bound C) - (W-bound C) > 0 & 2 |^ n > 0 ) by NEWTON:102, XREAL_1:52;
A4: (((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 A2, Lm2;
A5: (W-bound C) + ((p `1 ) - (W-bound C)) = p `1 ;
[\((((p `1 ) - (W-bound C)) / ((E-bound C) - (W-bound C))) * (2 |^ n))/] <= (((p `1 ) - (W-bound C)) / ((E-bound C) - (W-bound C))) * (2 |^ n) by INT_1:def 4;
then (((E-bound C) - (W-bound C)) / (2 |^ n)) * [\((((p `1 ) - (W-bound C)) / ((E-bound C) - (W-bound C))) * (2 |^ n))/] <= (((E-bound C) - (W-bound C)) / (2 |^ n)) * ((((p `1 ) - (W-bound C)) / ((E-bound C) - (W-bound C))) * (2 |^ n)) by A2, XREAL_1:66;
hence (W-bound C) + ((((E-bound C) - (W-bound C)) / (2 |^ n)) * (I - 2)) <= p `1 by A1, A4, A5, XREAL_1:8; :: thesis: verum