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); :: 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 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 } ; :: thesis: (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 :: according to XBOOLE_0:def 10 :: thesis: T c= (AffineMap (1,0,(1 / 2),(1 / 2))) .: S

proof

let x be object ; :: 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 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; :: thesis: verum

end;

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in T or x in (AffineMap (1,0,(1 / 2),(1 / 2))) .: S )
assume A8: x in T ; :: thesis: 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; :: thesis: verum

end;

hence (AffineMap (1,0,(1 / 2),(1 / 2))) .: S = T ; :: thesis: verum