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)
for O, I being Point of I[01] st a < b & c < d & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & a <= (f . O) `1 & (f . O) `1 <= b & c <= (f . I) `2 & (f . I) `2 <= d holds
( - 1 <= ((h * f) . O) `1 & ((h * f) . O) `1 <= 1 & - 1 <= ((h * f) . I) `2 & ((h * f) . I) `2 <= 1 )
let h be Function of (TOP-REAL 2),(TOP-REAL 2); :: thesis: for f being Function of I[01],(TOP-REAL 2)
for O, I being Point of I[01] st a < b & c < d & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & a <= (f . O) `1 & (f . O) `1 <= b & c <= (f . I) `2 & (f . I) `2 <= d holds
( - 1 <= ((h * f) . O) `1 & ((h * f) . O) `1 <= 1 & - 1 <= ((h * f) . I) `2 & ((h * f) . I) `2 <= 1 )
let f be Function of I[01],(TOP-REAL 2); :: thesis: for O, I being Point of I[01] st a < b & c < d & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & a <= (f . O) `1 & (f . O) `1 <= b & c <= (f . I) `2 & (f . I) `2 <= d holds
( - 1 <= ((h * f) . O) `1 & ((h * f) . O) `1 <= 1 & - 1 <= ((h * f) . I) `2 & ((h * f) . I) `2 <= 1 )
let O, I be Point of I[01]; :: thesis: ( a < b & c < d & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & a <= (f . O) `1 & (f . O) `1 <= b & c <= (f . I) `2 & (f . I) `2 <= d implies ( - 1 <= ((h * f) . O) `1 & ((h * f) . O) `1 <= 1 & - 1 <= ((h * f) . I) `2 & ((h * f) . I) `2 <= 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: a <= (f . O) `1 and
A5: (f . O) `1 <= b and
A6: c <= (f . I) `2 and
A7: (f . I) `2 <= d ; :: thesis: ( - 1 <= ((h * f) . O) `1 & ((h * f) . O) `1 <= 1 & - 1 <= ((h * f) . I) `2 & ((h * f) . I) `2 <= 1 )
A8: h . (f . O) = |[(((2 / (b - a)) * ((f . O) `1)) + (- ((b + a) / (b - a)))),(((2 / (d - c)) * ((f . O) `2)) + (- ((d + c) / (d - c))))]| by A3, JGRAPH_2:def 2;
A9: b - a > 0 by A1, XREAL_1:50;
then A10: 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 A9, XCMPLX_1:113
.= (b - a) * (((a + a) / (b - a)) / 2) by XCMPLX_1:82
.= ((b - a) * ((a + a) / (b - a))) / 2
.= (a + a) / 2 by A9, XCMPLX_1:87
.= a ;
then (2 / (b - a)) * (((- 1) - (- ((b + a) / (b - a)))) / (2 / (b - a))) <= (2 / (b - a)) * ((f . O) `1) by A4, A10, XREAL_1:64;
then (- 1) - (- ((b + a) / (b - a))) <= (2 / (b - a)) * ((f . O) `1) by A10, XCMPLX_1:87;
then A11: ((- 1) - (- ((b + a) / (b - a)))) + (- ((b + a) / (b - a))) <= ((2 / (b - a)) * ((f . O) `1)) + (- ((b + a) / (b - a))) by XREAL_1:6;
A12: d - c > 0 by A2, XREAL_1:50;
then A13: 2 / (d - c) > 0 by XREAL_1:139;
((- 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 A12, XCMPLX_1:113
.= (d - c) * (((c + c) / (d - c)) / 2) by XCMPLX_1:82
.= ((d - c) * ((c + c) / (d - c))) / 2
.= (c + c) / 2 by A12, XCMPLX_1:87
.= c ;
then (2 / (d - c)) * (((- 1) - (- ((d + c) / (d - c)))) / (2 / (d - c))) <= (2 / (d - c)) * ((f . I) `2) by A6, A13, XREAL_1:64;
then (- 1) - (- ((d + c) / (d - c))) <= (2 / (d - c)) * ((f . I) `2) by A13, XCMPLX_1:87;
then A14: ((- 1) - (- ((d + c) / (d - c)))) + (- ((d + c) / (d - c))) <= ((2 / (d - c)) * ((f . I) `2)) + (- ((d + c) / (d - c))) by XREAL_1:6;
A15: dom f = the carrier of I[01] by FUNCT_2:def 1;
then A16: (h * f) . I = h . (f . I) by FUNCT_1:13;
(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 A9, XCMPLX_1:113
.= (b - a) * (((b + b) / (b - a)) / 2) by XCMPLX_1:82
.= ((b - a) * ((b + b) / (b - a))) / 2
.= (b + b) / 2 by A9, XCMPLX_1:87
.= b ;
then (2 / (b - a)) * ((1 - (- ((b + a) / (b - a)))) / (2 / (b - a))) >= (2 / (b - a)) * ((f . O) `1) by A5, A10, XREAL_1:64;
then 1 - (- ((b + a) / (b - a))) >= (2 / (b - a)) * ((f . O) `1) by A10, XCMPLX_1:87;
then A17: (1 - (- ((b + a) / (b - a)))) + (- ((b + a) / (b - a))) >= ((2 / (b - a)) * ((f . O) `1)) + (- ((b + a) / (b - a))) by XREAL_1:6;
A18: h . (f . I) = |[(((2 / (b - a)) * ((f . I) `1)) + (- ((b + a) / (b - a)))),(((2 / (d - c)) * ((f . I) `2)) + (- ((d + c) / (d - c))))]| by A3, JGRAPH_2:def 2;
(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 A12, XCMPLX_1:113
.= (d - c) * (((d + d) / (d - c)) / 2) by XCMPLX_1:82
.= ((d - c) * ((d + d) / (d - c))) / 2
.= (d + d) / 2 by A12, XCMPLX_1:87
.= d ;
then (2 / (d - c)) * ((1 - (- ((d + c) / (d - c)))) / (2 / (d - c))) >= (2 / (d - c)) * ((f . I) `2) by A7, A13, XREAL_1:64;
then 1 - (- ((d + c) / (d - c))) >= (2 / (d - c)) * ((f . I) `2) by A13, XCMPLX_1:87;
then A19: (1 - (- ((d + c) / (d - c)))) + (- ((d + c) / (d - c))) >= ((2 / (d - c)) * ((f . I) `2)) + (- ((d + c) / (d - c))) by XREAL_1:6;
(h * f) . O = h . (f . O) by A15, FUNCT_1:13;
hence ( - 1 <= ((h * f) . O) `1 & ((h * f) . O) `1 <= 1 & - 1 <= ((h * f) . I) `2 & ((h * f) . I) `2 <= 1 ) by A16, A8, A18, A11, A17, A19, A14, EUCLID:52; :: thesis: verum