let S, T be Subset of (TOP-REAL 2); :: thesis: ( S = { p where p is Point of (TOP-REAL 2) : p `2 <= (2 * (p `1 )) - 1 } & T = { p where p is Point of (TOP-REAL 2) : p `2 <= p `1 } implies (AffineMap 1,0 ,(1 / 2),(1 / 2)) .: S = T )
assume A1:
( S = { p where p is Point of (TOP-REAL 2) : p `2 <= (2 * (p `1 )) - 1 } & T = { p where p is Point of (TOP-REAL 2) : p `2 <= p `1 } )
; :: thesis: (AffineMap 1,0 ,(1 / 2),(1 / 2)) .: S = T
set A = 1;
set B = 0 ;
set C = 1 / 2;
set D = 1 / 2;
set f = AffineMap 1,0 ,(1 / 2),(1 / 2);
(AffineMap 1,0 ,(1 / 2),(1 / 2)) .: S = T
proof
thus
(AffineMap 1,0 ,(1 / 2),(1 / 2)) .: S c= T
:: according to XBOOLE_0:def 10 :: thesis: T c= (AffineMap 1,0 ,(1 / 2),(1 / 2)) .: Sproof
let x be
set ;
:: according to TARSKI:def 3 :: thesis: ( not x in (AffineMap 1,0 ,(1 / 2),(1 / 2)) .: S or x in T )
assume
x in (AffineMap 1,0 ,(1 / 2),(1 / 2)) .: S
;
:: thesis: x in T
then consider y being
set such that A2:
(
y in dom (AffineMap 1,0 ,(1 / 2),(1 / 2)) &
y in S &
x = (AffineMap 1,0 ,(1 / 2),(1 / 2)) . y )
by FUNCT_1:def 12;
consider p being
Point of
(TOP-REAL 2) such that A3:
(
y = p &
p `2 <= (2 * (p `1 )) - 1 )
by A1, A2;
set b =
(AffineMap 1,0 ,(1 / 2),(1 / 2)) . p;
A4:
(AffineMap 1,0 ,(1 / 2),(1 / 2)) . p = |[((1 * (p `1 )) + 0 ),(((1 / 2) * (p `2 )) + (1 / 2))]|
by JGRAPH_2:def 2;
then A5:
((AffineMap 1,0 ,(1 / 2),(1 / 2)) . p) `1 = (1 * (p `1 )) + 0
by EUCLID:56;
A6:
((AffineMap 1,0 ,(1 / 2),(1 / 2)) . p) `2 = ((1 / 2) * (p `2 )) + (1 / 2)
by A4, EUCLID:56;
(1 / 2) * (p `2 ) <= (1 / 2) * ((2 * (p `1 )) - 1)
by A3, XREAL_1:66;
then
((1 / 2) * (p `2 )) + (1 / 2) <= ((p `1 ) - (1 / 2)) + (1 / 2)
by XREAL_1:8;
hence
x in T
by A1, A2, A3, A5, A6;
:: thesis: verum
end;
let x be
set ;
:: according to TARSKI:def 3 :: thesis: ( not x in T or x in (AffineMap 1,0 ,(1 / 2),(1 / 2)) .: S )
assume
x in T
;
:: thesis: x in (AffineMap 1,0 ,(1 / 2),(1 / 2)) .: S
then consider p being
Point of
(TOP-REAL 2) such that A7:
(
x = p &
p `2 <= p `1 )
by A1;
AffineMap 1,
0 ,
(1 / 2),
(1 / 2) is
onto
by JORDAN1K:36;
then
rng (AffineMap 1,0 ,(1 / 2),(1 / 2)) = the
carrier of
(TOP-REAL 2)
by FUNCT_2:def 3;
then consider y being
set such that A8:
(
y in dom (AffineMap 1,0 ,(1 / 2),(1 / 2)) &
x = (AffineMap 1,0 ,(1 / 2),(1 / 2)) . y )
by A7, FUNCT_1:def 5;
reconsider y =
y as
Point of
(TOP-REAL 2) by A8;
set b =
(AffineMap 1,0 ,(1 / 2),(1 / 2)) . y;
A9:
(AffineMap 1,0 ,(1 / 2),(1 / 2)) . y = |[((1 * (y `1 )) + 0 ),(((1 / 2) * (y `2 )) + (1 / 2))]|
by JGRAPH_2:def 2;
then A10:
((AffineMap 1,0 ,(1 / 2),(1 / 2)) . y) `1 = y `1
by EUCLID:56;
A11:
((AffineMap 1,0 ,(1 / 2),(1 / 2)) . y) `2 = ((1 / 2) * (y `2 )) + (1 / 2)
by A9, EUCLID:56;
2
* (((AffineMap 1,0 ,(1 / 2),(1 / 2)) . y) `2 ) <= 2
* (y `1 )
by A7, A8, A10, XREAL_1:66;
then
(2 * (((AffineMap 1,0 ,(1 / 2),(1 / 2)) . y) `2 )) - 1
<= (2 * (y `1 )) - 1
by XREAL_1:11;
then
y in S
by A1, A11;
hence
x in (AffineMap 1,0 ,(1 / 2),(1 / 2)) .: S
by A8, FUNCT_1:def 12;
:: thesis: verum
end;
hence
(AffineMap 1,0 ,(1 / 2),(1 / 2)) .: S = T
; :: thesis: verum