let a, b, c, d be Real; :: thesis: for h being Function of (TOP-REAL 2),(TOP-REAL 2)
for f being Function of I[01],(TOP-REAL 2) st a < b & c < d & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & rng f c= closed_inside_of_rectangle (a,b,c,d) holds
rng (h * f) c= closed_inside_of_rectangle ((- 1),1,(- 1),1)
let h be Function of (TOP-REAL 2),(TOP-REAL 2); :: thesis: for f being Function of I[01],(TOP-REAL 2) st a < b & c < d & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & rng f c= closed_inside_of_rectangle (a,b,c,d) holds
rng (h * f) c= closed_inside_of_rectangle ((- 1),1,(- 1),1)
let f be Function of I[01],(TOP-REAL 2); :: thesis: ( a < b & c < d & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & rng f c= closed_inside_of_rectangle (a,b,c,d) implies rng (h * f) c= closed_inside_of_rectangle ((- 1),1,(- 1),1) )
set A = 2 / (b - a);
set B = - ((b + a) / (b - a));
set C = 2 / (d - c);
set D = - ((d + c) / (d - c));
assume that
A1: a < b and
A2: c < d and
A3: h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) and
A4: rng f c= closed_inside_of_rectangle (a,b,c,d) ; :: thesis: rng (h * f) c= closed_inside_of_rectangle ((- 1),1,(- 1),1)
let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in rng (h * f) or x in closed_inside_of_rectangle ((- 1),1,(- 1),1) )
assume x in rng (h * f) ; :: thesis: x in closed_inside_of_rectangle ((- 1),1,(- 1),1)
then consider y being object such that
A5: y in dom (h * f) and
A6: x = (h * f) . y by FUNCT_1:def 3;
reconsider t0 = y as Point of I[01] by A5;
A7: (h * f) . t0 = h . (f . t0) by A5, FUNCT_1:12;
dom f = the carrier of I[01] by FUNCT_2:def 1;
then f . t0 in rng f by FUNCT_1:def 3;
then f . t0 in closed_inside_of_rectangle (a,b,c,d) by A4;
then f . t0 in { p where p is Point of (TOP-REAL 2) : ( a <= p `1 & p `1 <= b & c <= p `2 & p `2 <= d ) } by JGRAPH_6:def 2;
then A8: ex p being Point of (TOP-REAL 2) st
( f . t0 = p & a <= p `1 & p `1 <= b & c <= p `2 & p `2 <= d ) ;
reconsider p0 = x as Point of (TOP-REAL 2) by A5, A6, FUNCT_2:5;
A9: h . (f . t0) = |[(((2 / (b - a)) * ((f . t0) `1)) + (- ((b + a) / (b - a)))),(((2 / (d - c)) * ((f . t0) `2)) + (- ((d + c) / (d - c))))]| by A3, JGRAPH_2:def 2;
A10: b - a > 0 by A1, XREAL_1:50;
then A11: 2 / (b - a) > 0 by XREAL_1:139;
((- 1) - (- ((b + a) / (b - a)))) / (2 / (b - a)) = ((- 1) + ((b + a) / (b - a))) / (2 / (b - a))
.= ((((- 1) * (b - a)) + (b + a)) / (b - a)) / (2 / (b - a)) by A10, XCMPLX_1:113
.= (((a + a) / (b - a)) / 2) * (b - a) by XCMPLX_1:82
.= ((b - a) * ((a + a) / (b - a))) / 2
.= (a + a) / 2 by A10, XCMPLX_1:87
.= a ;
then (2 / (b - a)) * (((- 1) - (- ((b + a) / (b - a)))) / (2 / (b - a))) <= (2 / (b - a)) * ((f . t0) `1) by A11, A8, XREAL_1:64;
then (- 1) - (- ((b + a) / (b - a))) <= (2 / (b - a)) * ((f . t0) `1) by A11, XCMPLX_1:87;
then ((- 1) - (- ((b + a) / (b - a)))) + (- ((b + a) / (b - a))) <= ((2 / (b - a)) * ((f . t0) `1)) + (- ((b + a) / (b - a))) by XREAL_1:6;
then A12: - 1 <= p0 `1 by A6, A9, A7, EUCLID:52;
A13: d - c > 0 by A2, XREAL_1:50;
then A14: 2 / (d - c) > 0 by XREAL_1:139;
(1 - (- ((b + a) / (b - a)))) / (2 / (b - a)) = (1 + ((b + a) / (b - a))) / (2 / (b - a))
.= (((1 * (b - a)) + (b + a)) / (b - a)) / (2 / (b - a)) by A10, XCMPLX_1:113
.= (((b + b) / (b - a)) / 2) * (b - a) by XCMPLX_1:82
.= ((b - a) * ((b + b) / (b - a))) / 2
.= (b + b) / 2 by A10, XCMPLX_1:87
.= b ;
then (2 / (b - a)) * ((1 - (- ((b + a) / (b - a)))) / (2 / (b - a))) >= (2 / (b - a)) * ((f . t0) `1) by A11, A8, XREAL_1:64;
then 1 - (- ((b + a) / (b - a))) >= (2 / (b - a)) * ((f . t0) `1) by A11, XCMPLX_1:87;
then (1 - (- ((b + a) / (b - a)))) + (- ((b + a) / (b - a))) >= ((2 / (b - a)) * ((f . t0) `1)) + (- ((b + a) / (b - a))) by XREAL_1:6;
then A15: p0 `1 <= 1 by A6, A9, A7, EUCLID:52;
(1 - (- ((d + c) / (d - c)))) / (2 / (d - c)) = (1 + ((d + c) / (d - c))) / (2 / (d - c))
.= (((1 * (d - c)) + (d + c)) / (d - c)) / (2 / (d - c)) by A13, XCMPLX_1:113
.= (((d + d) / (d - c)) / 2) * (d - c) by XCMPLX_1:82
.= ((d - c) * ((d + d) / (d - c))) / 2
.= (d + d) / 2 by A13, XCMPLX_1:87
.= d ;
then (2 / (d - c)) * ((1 - (- ((d + c) / (d - c)))) / (2 / (d - c))) >= (2 / (d - c)) * ((f . t0) `2) by A14, A8, XREAL_1:64;
then 1 - (- ((d + c) / (d - c))) >= (2 / (d - c)) * ((f . t0) `2) by A14, XCMPLX_1:87;
then (1 - (- ((d + c) / (d - c)))) + (- ((d + c) / (d - c))) >= ((2 / (d - c)) * ((f . t0) `2)) + (- ((d + c) / (d - c))) by XREAL_1:6;
then A16: p0 `2 <= 1 by A6, A9, A7, EUCLID:52;
((- 1) - (- ((d + c) / (d - c)))) / (2 / (d - c)) = ((- 1) + ((d + c) / (d - c))) / (2 / (d - c))
.= ((((- 1) * (d - c)) + (d + c)) / (d - c)) / (2 / (d - c)) by A13, XCMPLX_1:113
.= (((c + c) / (d - c)) / 2) * (d - c) by XCMPLX_1:82
.= ((d - c) * ((c + c) / (d - c))) / 2
.= (c + c) / 2 by A13, XCMPLX_1:87
.= c ;
then (2 / (d - c)) * (((- 1) - (- ((d + c) / (d - c)))) / (2 / (d - c))) <= (2 / (d - c)) * ((f . t0) `2) by A14, A8, XREAL_1:64;
then (- 1) - (- ((d + c) / (d - c))) <= (2 / (d - c)) * ((f . t0) `2) by A14, XCMPLX_1:87;
then ((- 1) - (- ((d + c) / (d - c)))) + (- ((d + c) / (d - c))) <= ((2 / (d - c)) * ((f . t0) `2)) + (- ((d + c) / (d - c))) by XREAL_1:6;
then - 1 <= p0 `2 by A6, A9, A7, EUCLID:52;
then x in { p2 where p2 is Point of (TOP-REAL 2) : ( - 1 <= p2 `1 & p2 `1 <= 1 & - 1 <= p2 `2 & p2 `2 <= 1 ) } by A16, A12, A15;
hence x in closed_inside_of_rectangle ((- 1),1,(- 1),1) by JGRAPH_6:def 2; :: thesis: verum