let a, b, c, d be real number ; :: 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))) & c <= (f . O) `2 & (f . O) `2 < (f . I) `2 & (f . I) `2 <= d holds
( - 1 <= ((h * f) . O) `2 & ((h * f) . O) `2 < ((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))) & c <= (f . O) `2 & (f . O) `2 < (f . I) `2 & (f . I) `2 <= d holds
( - 1 <= ((h * f) . O) `2 & ((h * f) . O) `2 < ((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))) & c <= (f . O) `2 & (f . O) `2 < (f . I) `2 & (f . I) `2 <= d holds
( - 1 <= ((h * f) . O) `2 & ((h * f) . O) `2 < ((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))) & c <= (f . O) `2 & (f . O) `2 < (f . I) `2 & (f . I) `2 <= d implies ( - 1 <= ((h * f) . O) `2 & ((h * f) . O) `2 < ((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 A1: ( a < b & c < d & h = AffineMap (2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))) & c <= (f . O) `2 & (f . O) `2 < (f . I) `2 & (f . I) `2 <= d ) ; :: thesis: ( - 1 <= ((h * f) . O) `2 & ((h * f) . O) `2 < ((h * f) . I) `2 & ((h * f) . I) `2 <= 1 )
then A2: d - c > 0 by XREAL_1:52;
then A3: 2 / (d - c) > 0 by XREAL_1:141;
A4: dom f = the carrier of I[01] by FUNCT_2:def 1;
then A5: (h * f) . O = h . (f . O) by FUNCT_1:23;
A6: (h * f) . I = h . (f . I) by A4, FUNCT_1:23;
A7: h . (f . O) = |[(((2 / (b - a)) * ((f . O) `1 )) + (- ((b + a) / (b - a)))),(((2 / (d - c)) * ((f . O) `2 )) + (- ((d + c) / (d - c))))]| by A1, JGRAPH_2:def 2;
A8: h . (f . I) = |[(((2 / (b - a)) * ((f . I) `1 )) + (- ((b + a) / (b - a)))),(((2 / (d - c)) * ((f . I) `2 )) + (- ((d + c) / (d - c))))]| by A1, 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 A2, XCMPLX_1:114
.= (((c + c) / (d - c)) / 2) * (d - c) by XCMPLX_1:83
.= ((d - c) * ((c + c) / (d - c))) / 2
.= (c + c) / 2 by A2, XCMPLX_1:88
.= c ;
then (2 / (d - c)) * (((- 1) - (- ((d + c) / (d - c)))) / (2 / (d - c))) <= (2 / (d - c)) * ((f . O) `2 ) by A1, A3, XREAL_1:66;
then (- 1) - (- ((d + c) / (d - c))) <= (2 / (d - c)) * ((f . O) `2 ) by A3, XCMPLX_1:88;
then A9: ((- 1) - (- ((d + c) / (d - c)))) + (- ((d + c) / (d - c))) <= ((2 / (d - c)) * ((f . O) `2 )) + (- ((d + c) / (d - c))) by XREAL_1:8;
(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 A2, XCMPLX_1:114
.= (((d + d) / (d - c)) / 2) * (d - c) by XCMPLX_1:83
.= ((d - c) * ((d + d) / (d - c))) / 2
.= (d + d) / 2 by A2, XCMPLX_1:88
.= d ;
then (2 / (d - c)) * ((1 - (- ((d + c) / (d - c)))) / (2 / (d - c))) >= (2 / (d - c)) * ((f . I) `2 ) by A1, A3, XREAL_1:66;
then 1 - (- ((d + c) / (d - c))) >= (2 / (d - c)) * ((f . I) `2 ) by A3, XCMPLX_1:88;
then A10: (1 - (- ((d + c) / (d - c)))) + (- ((d + c) / (d - c))) >= ((2 / (d - c)) * ((f . I) `2 )) + (- ((d + c) / (d - c))) by XREAL_1:8;
(2 / (d - c)) * ((f . O) `2 ) < (2 / (d - c)) * ((f . I) `2 ) by A1, A3, XREAL_1:70;
then ((2 / (d - c)) * ((f . O) `2 )) + (- ((d + c) / (d - c))) < ((2 / (d - c)) * ((f . I) `2 )) + (- ((d + c) / (d - c))) by XREAL_1:10;
then ((2 / (d - c)) * ((f . O) `2 )) + (- ((d + c) / (d - c))) < ((h * f) . I) `2 by A6, A8, EUCLID:56;
hence ( - 1 <= ((h * f) . O) `2 & ((h * f) . O) `2 < ((h * f) . I) `2 & ((h * f) . I) `2 <= 1 ) by A5, A6, A7, A8, A9, A10, EUCLID:56; :: thesis: verum