set f = AffineMap (1,0,(1 / 2),(1 / 2));
set A = 1;
set B = 0 ;
set C = 1 / 2;
set D = 1 / 2;
let S, T be Subset of (TOP-REAL 2); ( 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 that
A1:
S = { p where p is Point of (TOP-REAL 2) : p `2 <= (2 * (p `1)) - 1 }
and
A2:
T = { p where p is Point of (TOP-REAL 2) : p `2 <= p `1 }
; (AffineMap (1,0,(1 / 2),(1 / 2))) .: S = T
(AffineMap (1,0,(1 / 2),(1 / 2))) .: S = T
proof
thus
(AffineMap (1,0,(1 / 2),(1 / 2))) .: S c= T
XBOOLE_0:def 10 T c= (AffineMap (1,0,(1 / 2),(1 / 2))) .: Sproof
let x be
object ;
TARSKI:def 3 ( 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
;
x in T
then consider y being
object such that
y in dom (AffineMap (1,0,(1 / 2),(1 / 2)))
and A3:
y in S
and A4:
x = (AffineMap (1,0,(1 / 2),(1 / 2))) . y
by FUNCT_1:def 6;
consider p being
Point of
(TOP-REAL 2) such that A5:
y = p
and A6:
p `2 <= (2 * (p `1)) - 1
by A1, A3;
set b =
(AffineMap (1,0,(1 / 2),(1 / 2))) . p;
(AffineMap (1,0,(1 / 2),(1 / 2))) . p = |[((1 * (p `1)) + 0),(((1 / 2) * (p `2)) + (1 / 2))]|
by JGRAPH_2:def 2;
then A7:
(
((AffineMap (1,0,(1 / 2),(1 / 2))) . p) `1 = (1 * (p `1)) + 0 &
((AffineMap (1,0,(1 / 2),(1 / 2))) . p) `2 = ((1 / 2) * (p `2)) + (1 / 2) )
by EUCLID:52;
(1 / 2) * (p `2) <= (1 / 2) * ((2 * (p `1)) - 1)
by A6, XREAL_1:64;
then
((1 / 2) * (p `2)) + (1 / 2) <= ((p `1) - (1 / 2)) + (1 / 2)
by XREAL_1:6;
hence
x in T
by A2, A4, A5, A7;
verum
end;
let x be
object ;
TARSKI:def 3 ( not x in T or x in (AffineMap (1,0,(1 / 2),(1 / 2))) .: S )
assume A8:
x in T
;
x in (AffineMap (1,0,(1 / 2),(1 / 2))) .: S
then A9:
ex
p being
Point of
(TOP-REAL 2) st
(
x = p &
p `2 <= p `1 )
by A2;
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
object such that A10:
y in dom (AffineMap (1,0,(1 / 2),(1 / 2)))
and A11:
x = (AffineMap (1,0,(1 / 2),(1 / 2))) . y
by A8, FUNCT_1:def 3;
reconsider y =
y as
Point of
(TOP-REAL 2) by A10;
set b =
(AffineMap (1,0,(1 / 2),(1 / 2))) . y;
A12:
(AffineMap (1,0,(1 / 2),(1 / 2))) . y = |[((1 * (y `1)) + 0),(((1 / 2) * (y `2)) + (1 / 2))]|
by JGRAPH_2:def 2;
then
((AffineMap (1,0,(1 / 2),(1 / 2))) . y) `1 = y `1
by EUCLID:52;
then
2
* (((AffineMap (1,0,(1 / 2),(1 / 2))) . y) `2) <= 2
* (y `1)
by A9, A11, XREAL_1:64;
then A13:
(2 * (((AffineMap (1,0,(1 / 2),(1 / 2))) . y) `2)) - 1
<= (2 * (y `1)) - 1
by XREAL_1:9;
((AffineMap (1,0,(1 / 2),(1 / 2))) . y) `2 = ((1 / 2) * (y `2)) + (1 / 2)
by A12, EUCLID:52;
then
y in S
by A1, A13;
hence
x in (AffineMap (1,0,(1 / 2),(1 / 2))) .: S
by A10, A11, FUNCT_1:def 6;
verum
end;
hence
(AffineMap (1,0,(1 / 2),(1 / 2))) .: S = T
; verum