let p1, p2, p3, p4 be Point of (TOP-REAL 2); :: thesis:
let a, b, c, d be real number ; :: thesis:
let f, g be Function of I[01] ,(TOP-REAL 2); :: thesis:
assume that
A1: a < b and
A2: c < d and
A3: p1 `2 = c and
A4: p2 `2 = c and
A5: p3 `2 = c and
A6: p4 `2 = c and
A7: b >= p1 `1 and
A8: p1 `1 > p2 `1 and
A9: p2 `1 > p3 `1 and
A10: p3 `1 > p4 `1 and
A11: p4 `1 > a and
A12: f . 0 = p1 and
A13: f . 1 = p3 and
A14: g . 0 = p2 and
A15: g . 1 = p4 and
A16: ( f is continuous & f is one-to-one ) and
A17: ( g is continuous & g is one-to-one ) and
A18: rng f c= closed_inside_of_rectangle a,b,c,d and
A19: rng g c= closed_inside_of_rectangle a,b,c,d ; :: thesis:
set A = 2 / (b - a);
set B = - ((b + a) / (b - a));
set C = 2 / (d - c);
set D = - ((d + c) / (d - c));
b - a > 0 by A1, XREAL_1:52;
then A20: 2 / (b - a) > 0 by XREAL_1:141;
d - c > 0 by A2, XREAL_1:52;
then A21: 2 / (d - c) > 0 by XREAL_1:141;
set h = AffineMap (2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)));
A22: p1 `1 > p3 `1 by A8, A9, XXREAL_0:2;
A23: p3 `1 > a by A10, A11, XXREAL_0:2;
A24: p2 `1 > p4 `1 by A9, A10, XXREAL_0:2;
A25: b > p2 `1 by A7, A8, XXREAL_0:2;
A26: ( AffineMap (2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))) is being_homeomorphism & ( for p11, p21 being Point of (TOP-REAL 2) st p11 `2 < p21 `2 holds
((AffineMap (2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) . p11) `2 < ((AffineMap (2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) . p21) `2 ) ) by A20, A21, Th51;
A27: ((AffineMap (2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) . p1) `1 > ((AffineMap (2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) . p2) `1 by A8, A20, A21, Th50;
A28: ((AffineMap (2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) . p2) `1 > ((AffineMap (2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) . p3) `1 by A9, A20, A21, Th50;
A29: ((AffineMap (2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) . p3) `1 > ((AffineMap (2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) . p4) `1 by A10, A20, A21, Th50;
reconsider f2 = (AffineMap (2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) * f as Function of I[01] ,(TOP-REAL 2) ;
reconsider g2 = (AffineMap (2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) * g as Function of I[01] ,(TOP-REAL 2) ;
reconsider O = 0 , I = 1 as Point of I[01] by BORSUK_1:83, XXREAL_1:1;
A30: dom f = the carrier of I[01] by FUNCT_2:def 1;
A31: dom g = the carrier of I[01] by FUNCT_2:def 1;
A32: (AffineMap (2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) . p1 = f2 . O by A12, A30, FUNCT_1:23;
A33: (AffineMap (2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) . p2 = g2 . O by A14, A31, FUNCT_1:23;
A34: (AffineMap (2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) . p3 = f2 . I by A13, A30, FUNCT_1:23;
A35: (AffineMap (2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) . p4 = g2 . I by A15, A31, FUNCT_1:23;
A36: ( (f . O) `2 = c & (f . I) `2 = c ) by A3, A5, A12, A13;
A37: ( f2 is continuous & f2 is one-to-one ) by A1, A2, A16, Th53;
A38: ( (f2 . O) `2 = - 1 & (f2 . I) `2 = - 1 ) by A1, A2, A3, A5, A12, A13, Th57;
A39: ( 1 >= (f2 . O) `1 & (f2 . O) `1 > (f2 . I) `1 & (f2 . I) `1 > - 1 ) by A1, A2, A7, A12, A13, A22, A23, A36, Th67;
A40: rng f2 c= closed_inside_of_rectangle (- 1),1,(- 1),1 by A1, A2, A18, Th52;
A41: ( (g . O) `2 = c & (g . I) `2 = c ) by A4, A6, A14, A15;
A42: ( g2 is continuous & g2 is one-to-one ) by A1, A2, A17, Th53;
A43: ( (g2 . O) `2 = - 1 & (g2 . I) `2 = - 1 ) by A1, A2, A4, A6, A14, A15, Th57;
A44: ( 1 >= (g2 . O) `1 & (g2 . O) `1 > (g2 . I) `1 & (g2 . I) `1 > - 1 ) by A1, A2, A11, A14, A15, A24, A25, A41, Th67;
X: rng g2 c= closed_inside_of_rectangle (- 1),1,(- 1),1 by A1, A2, A19, Th52;
then f2 . O,g2 . O,f2 . I,g2 . I are_in_this_order_on rectangle (- 1),1,(- 1),1 by A27, A28, A29, A32, A33, A34, A35, A37, A38, A39, A40, A42, A43, A44, Th48;
then rng f2 meets rng g2 by A27, A28, A29, A32, A33, A34, A35, A37, A38, A39, A40, A42, A43, A44, JGRAPH_6:89, X;
then A45: (rng f2) /\ (rng g2) <> {} by XBOOLE_0:def 7;
consider x being Element of (rng f2) /\ (rng g2);
A46: ( x in rng f2 & x in rng g2 ) by A45, XBOOLE_0:def 4;
then consider z1 being set such that
A47: ( z1 in dom f2 & x = f2 . z1 ) by FUNCT_1:def 5;
consider z2 being set such that
A48: ( z2 in dom g2 & x = g2 . z2 ) by A46, FUNCT_1:def 5;
A49: x = (AffineMap (2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) . (f . z1) by A30, A47, FUNCT_1:23;
A50: x = (AffineMap (2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) . (g . z2) by A31, A48, FUNCT_1:23;
A51: f . z1 in rng f by A30, A47, FUNCT_1:def 5;
A52: g . z2 in rng g by A31, A48, FUNCT_1:def 5;
f . z1 in the carrier of (TOP-REAL 2) by A47, FUNCT_2:7;
then A53: f . z1 in dom (AffineMap (2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) by FUNCT_2:def 1;
g . z2 in the carrier of (TOP-REAL 2) by A48, FUNCT_2:7;
then A54: g . z2 in dom (AffineMap (2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) by FUNCT_2:def 1;
AffineMap (2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))) is one-to-one by A26, TOPS_2:def 5;
then f . z1 = g . z2 by A49, A50, A53, A54, FUNCT_1:def 8;
hence rng f meets rng g by A51, A52, XBOOLE_0:3; :: thesis: