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 <= 1 - (2 * (p `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 <= 1 - (2 * (p `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 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
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 12;
consider p being Point of (TOP-REAL 2) such that
A5: y = p and
A6: p `2 <= 1 - (2 * (p `1 )) by A1, A3;
set b = (AffineMap 1,0 ,(1 / 2),(- (1 / 2))) . p;
(1 / 2) * (p `2 ) <= (1 / 2) * (1 - (2 * (p `1 ))) by A6, XREAL_1:66;
then A7: ((1 / 2) * (p `2 )) + (- (1 / 2)) <= ((1 / 2) - (p `1 )) + (- (1 / 2)) by XREAL_1:8;
A8: (AffineMap 1,0 ,(1 / 2),(- (1 / 2))) . p = |[((1 * (p `1 )) + 0 ),(((1 / 2) * (p `2 )) + (- (1 / 2)))]| by JGRAPH_2:def 2;
then ((AffineMap 1,0 ,(1 / 2),(- (1 / 2))) . p) `1 = (1 * (p `1 )) + 0 by EUCLID:56;
then ((AffineMap 1,0 ,(1 / 2),(- (1 / 2))) . p) `2 <= - (((AffineMap 1,0 ,(1 / 2),(- (1 / 2))) . p) `1 ) by A8, A7, EUCLID:56;
hence x in T by A2, A4, A5; :: 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 A9: x in T ; :: thesis: x in (AffineMap 1,0 ,(1 / 2),(- (1 / 2))) .: S
then A10: 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 set such that
A11: y in dom (AffineMap 1,0 ,(1 / 2),(- (1 / 2))) and
A12: x = (AffineMap 1,0 ,(1 / 2),(- (1 / 2))) . y by A9, FUNCT_1:def 5;
reconsider y = y as Point of (TOP-REAL 2) by A11;
set b = (AffineMap 1,0 ,(1 / 2),(- (1 / 2))) . y;
A13: (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:56;
then 2 * (((AffineMap 1,0 ,(1 / 2),(- (1 / 2))) . y) `2 ) <= 2 * (- (y `1 )) by A10, A12, XREAL_1:66;
then A14: (2 * (((AffineMap 1,0 ,(1 / 2),(- (1 / 2))) . y) `2 )) + 1 <= (2 * (- (y `1 ))) + 1 by XREAL_1:8;
((AffineMap 1,0 ,(1 / 2),(- (1 / 2))) . y) `2 = ((1 / 2) * (y `2 )) + (- (1 / 2)) by A13, EUCLID:56;
then y `2 <= 1 - (2 * (y `1 )) by A14;
then y in S by A1;
hence x in (AffineMap 1,0 ,(1 / 2),(- (1 / 2))) .: S by A11, A12, FUNCT_1:def 12; :: thesis: verum

end;

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