let C be Simple_closed_curve; :: thesis: for n being Nat
for p being Point of (TOP-REAL 2)
for J being Integer st J = [\(((((p `2) - (S-bound C)) / ((N-bound C) - (S-bound C))) * (2 |^ n)) + 2)/] holds
(S-bound C) + ((((N-bound C) - (S-bound C)) / (2 |^ n)) * (J - 2)) <= p `2
let n be Nat; :: thesis: for p being Point of (TOP-REAL 2)
for J being Integer st J = [\(((((p `2) - (S-bound C)) / ((N-bound C) - (S-bound C))) * (2 |^ n)) + 2)/] holds
(S-bound C) + ((((N-bound C) - (S-bound C)) / (2 |^ n)) * (J - 2)) <= p `2
let p be Point of (TOP-REAL 2); :: thesis: for J being Integer st J = [\(((((p `2) - (S-bound C)) / ((N-bound C) - (S-bound C))) * (2 |^ n)) + 2)/] holds
(S-bound C) + ((((N-bound C) - (S-bound C)) / (2 |^ n)) * (J - 2)) <= p `2
set W = S-bound C;
set EW = (N-bound C) - (S-bound C);
set PW = (p `2) - (S-bound C);
set KI = [\((((p `2) - (S-bound C)) / ((N-bound C) - (S-bound C))) * (2 |^ n))/];
let I be Integer; :: thesis: ( I = [\(((((p `2) - (S-bound C)) / ((N-bound C) - (S-bound C))) * (2 |^ n)) + 2)/] implies (S-bound C) + ((((N-bound C) - (S-bound C)) / (2 |^ n)) * (I - 2)) <= p `2 )
A1: (N-bound C) - (S-bound C) > 0 by TOPREAL5:16, XREAL_1:50;
2 |^ n > 0 by NEWTON:83;
then A2: (((N-bound C) - (S-bound C)) / (2 |^ n)) * ((((p `2) - (S-bound C)) / ((N-bound C) - (S-bound C))) * (2 |^ n)) = (p `2) - (S-bound C) by A1, Lm2;
assume I = [\(((((p `2) - (S-bound C)) / ((N-bound C) - (S-bound C))) * (2 |^ n)) + 2)/] ; :: thesis: (S-bound C) + ((((N-bound C) - (S-bound C)) / (2 |^ n)) * (I - 2)) <= p `2
then A3: I - 2 = [\((((p `2) - (S-bound C)) / ((N-bound C) - (S-bound C))) * (2 |^ n))/] by Lm1;
[\((((p `2) - (S-bound C)) / ((N-bound C) - (S-bound C))) * (2 |^ n))/] <= (((p `2) - (S-bound C)) / ((N-bound C) - (S-bound C))) * (2 |^ n) by INT_1:def 6;
then A4: (((N-bound C) - (S-bound C)) / (2 |^ n)) * [\((((p `2) - (S-bound C)) / ((N-bound C) - (S-bound C))) * (2 |^ n))/] <= (((N-bound C) - (S-bound C)) / (2 |^ n)) * ((((p `2) - (S-bound C)) / ((N-bound C) - (S-bound C))) * (2 |^ n)) by A1, XREAL_1:64;
(S-bound C) + ((p `2) - (S-bound C)) = p `2 ;
hence (S-bound C) + ((((N-bound C) - (S-bound C)) / (2 |^ n)) * (I - 2)) <= p `2 by A3, A2, A4, XREAL_1:6; :: thesis: verum