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 & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & a <= (f . O) `1 & (f . O) `1 <= b & a < (f . I) `1 & (f . I) `1 <= b holds
( - 1 <= ((h * f) . O) `1 & ((h * f) . O) `1 <= 1 & - 1 < ((h * f) . I) `1 & ((h * f) . I) `1 <= 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 & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & a <= (f . O) `1 & (f . O) `1 <= b & a < (f . I) `1 & (f . I) `1 <= b holds
( - 1 <= ((h * f) . O) `1 & ((h * f) . O) `1 <= 1 & - 1 < ((h * f) . I) `1 & ((h * f) . I) `1 <= 1 )
let f be Function of I[01],(TOP-REAL 2); :: thesis: for O, I being Point of I[01] st a < b & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & a <= (f . O) `1 & (f . O) `1 <= b & a < (f . I) `1 & (f . I) `1 <= b holds
( - 1 <= ((h * f) . O) `1 & ((h * f) . O) `1 <= 1 & - 1 < ((h * f) . I) `1 & ((h * f) . I) `1 <= 1 )
let O, I be Point of I[01]; :: thesis: ( a < b & h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) & a <= (f . O) `1 & (f . O) `1 <= b & a < (f . I) `1 & (f . I) `1 <= b implies ( - 1 <= ((h * f) . O) `1 & ((h * f) . O) `1 <= 1 & - 1 < ((h * f) . I) `1 & ((h * f) . I) `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: h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) and
A3: a <= (f . O) `1 and
A4: (f . O) `1 <= b and
A5: a < (f . I) `1 and
A6: (f . I) `1 <= b ; :: thesis: ( - 1 <= ((h * f) . O) `1 & ((h * f) . O) `1 <= 1 & - 1 < ((h * f) . I) `1 & ((h * f) . I) `1 <= 1 )
A7: b - a > 0 by A1, XREAL_1:50;
then A8: 2 / (b - a) > 0 by XREAL_1:139;
A9: (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 A7, 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 A7, XCMPLX_1:87
.= b ;
then (2 / (b - a)) * ((1 - (- ((b + a) / (b - a)))) / (2 / (b - a))) >= (2 / (b - a)) * ((f . O) `1) by A4, A8, XREAL_1:64;
then 1 - (- ((b + a) / (b - a))) >= (2 / (b - a)) * ((f . O) `1) by A8, XCMPLX_1:87;
then A10: (1 - (- ((b + a) / (b - a)))) + (- ((b + a) / (b - a))) >= ((2 / (b - a)) * ((f . O) `1)) + (- ((b + a) / (b - a))) by XREAL_1:6;
(2 / (b - a)) * ((1 - (- ((b + a) / (b - a)))) / (2 / (b - a))) >= (2 / (b - a)) * ((f . I) `1) by A6, A8, A9, XREAL_1:64;
then 1 - (- ((b + a) / (b - a))) >= (2 / (b - a)) * ((f . I) `1) by A8, XCMPLX_1:87;
then A11: (1 - (- ((b + a) / (b - a)))) + (- ((b + a) / (b - a))) >= ((2 / (b - a)) * ((f . I) `1)) + (- ((b + a) / (b - a))) by XREAL_1:6;
A12: h . (f . I) = |[(((2 / (b - a)) * ((f . I) `1)) + (- ((b + a) / (b - a)))),(((2 / (d - c)) * ((f . I) `2)) + (- ((d + c) / (d - c))))]| by A2, JGRAPH_2:def 2;
A13: h . (f . O) = |[(((2 / (b - a)) * ((f . O) `1)) + (- ((b + a) / (b - a)))),(((2 / (d - c)) * ((f . O) `2)) + (- ((d + c) / (d - c))))]| by A2, JGRAPH_2:def 2;
A14: dom f = the carrier of I[01] by FUNCT_2:def 1;
then A15: (h * f) . I = h . (f . I) by FUNCT_1:13;
A16: ((- 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 A7, 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 A7, XCMPLX_1:87
.= a ;
then (2 / (b - a)) * (((- 1) - (- ((b + a) / (b - a)))) / (2 / (b - a))) <= (2 / (b - a)) * ((f . O) `1) by A3, A8, XREAL_1:64;
then (- 1) - (- ((b + a) / (b - a))) <= (2 / (b - a)) * ((f . O) `1) by A8, 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;
(2 / (b - a)) * (((- 1) - (- ((b + a) / (b - a)))) / (2 / (b - a))) < (2 / (b - a)) * ((f . I) `1) by A5, A8, A16, XREAL_1:68;
then (- 1) - (- ((b + a) / (b - a))) < (2 / (b - a)) * ((f . I) `1) by A8, XCMPLX_1:87;
then A18: ((- 1) - (- ((b + a) / (b - a)))) + (- ((b + a) / (b - a))) < ((2 / (b - a)) * ((f . I) `1)) + (- ((b + a) / (b - a))) by XREAL_1:8;
(h * f) . O = h . (f . O) by A14, FUNCT_1:13;
hence ( - 1 <= ((h * f) . O) `1 & ((h * f) . O) `1 <= 1 & - 1 < ((h * f) . I) `1 & ((h * f) . I) `1 <= 1 ) by A15, A13, A12, A17, A10, A11, A18, EUCLID:52; :: thesis: verum