let p1, p2, p3, p4 be Point of (TOP-REAL 2); :: thesis: for a, b, c, d being real number
for f, g being Function of I[01] ,(TOP-REAL 2) st a < b & c < d & p1 `2 = d & p2 `2 = d & p3 `1 = b & p4 `1 = b & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 <= b & c <= p4 `2 & p4 `2 < p3 `2 & p3 `2 <= d & f . 0 = p1 & f . 1 = p3 & g . 0 = p2 & g . 1 = p4 & f is continuous & f is one-to-one & g is continuous & g is one-to-one & rng f c= closed_inside_of_rectangle a,b,c,d & rng g c= closed_inside_of_rectangle a,b,c,d holds
rng f meets rng g
let a, b, c, d be real number ; :: thesis: for f, g being Function of I[01] ,(TOP-REAL 2) st a < b & c < d & p1 `2 = d & p2 `2 = d & p3 `1 = b & p4 `1 = b & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 <= b & c <= p4 `2 & p4 `2 < p3 `2 & p3 `2 <= d & f . 0 = p1 & f . 1 = p3 & g . 0 = p2 & g . 1 = p4 & f is continuous & f is one-to-one & g is continuous & g is one-to-one & rng f c= closed_inside_of_rectangle a,b,c,d & rng g c= closed_inside_of_rectangle a,b,c,d holds
rng f meets rng g
let f, g be Function of I[01] ,(TOP-REAL 2); :: thesis: ( a < b & c < d & p1 `2 = d & p2 `2 = d & p3 `1 = b & p4 `1 = b & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 <= b & c <= p4 `2 & p4 `2 < p3 `2 & p3 `2 <= d & f . 0 = p1 & f . 1 = p3 & g . 0 = p2 & g . 1 = p4 & f is continuous & f is one-to-one & g is continuous & g is one-to-one & rng f c= closed_inside_of_rectangle a,b,c,d & rng g c= closed_inside_of_rectangle a,b,c,d implies rng f meets rng g )
assume A1: ( a < b & c < d & p1 `2 = d & p2 `2 = d & p3 `1 = b & p4 `1 = b & a <= p1 `1 & p1 `1 < p2 `1 & p2 `1 <= b & c <= p4 `2 & p4 `2 < p3 `2 & p3 `2 <= d & f . 0 = p1 & f . 1 = p3 & g . 0 = p2 & g . 1 = p4 & f is continuous & f is one-to-one & g is continuous & g is one-to-one & rng f c= closed_inside_of_rectangle a,b,c,d & rng g c= closed_inside_of_rectangle a,b,c,d ) ; :: thesis: rng f meets rng g
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 A2: 2 / (b - a) > 0 by XREAL_1:141;
d - c > 0 by A1, XREAL_1:52;
then A3: 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)));
A4: d > p4 `2 by A1, XXREAL_0:2;
A5: p1 `1 <= b by A1, XXREAL_0:2;
A6: p3 `2 > c by A1, XXREAL_0:2;
A7: a < p2 `1 by A1, XXREAL_0:2;
A8: ( 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 A2, A3, Th51;
A9: ((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 A1, A2, A3, Th50;
A10: ((AffineMap (2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) . p3) `2 > ((AffineMap (2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) . p4) `2 by A1, A2, A3, Th51;
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;
A11: dom f = the carrier of I[01] by FUNCT_2:def 1;
A12: dom g = the carrier of I[01] by FUNCT_2:def 1;
A13: (AffineMap (2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) . p1 = f2 . O by A1, A11, FUNCT_1:23;
A14: (AffineMap (2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) . p2 = g2 . O by A1, A12, FUNCT_1:23;
A15: (AffineMap (2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) . p3 = f2 . I by A1, A11, FUNCT_1:23;
A16: (AffineMap (2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) . p4 = g2 . I by A1, A12, FUNCT_1:23;
A17: ( f2 is continuous & f2 is one-to-one ) by A1, Th53;
A18: ( (f2 . O) `2 = 1 & (f2 . I) `1 = 1 ) by A1, Th55, Th56;
A19: ( - 1 <= (f2 . O) `1 & (f2 . O) `1 <= 1 & - 1 <= (f2 . I) `2 & (f2 . I) `2 <= 1 ) by A1, A5, A6, Th63;
A20: rng f2 c= closed_inside_of_rectangle (- 1),1,(- 1),1 by A1, Th52;
A21: ( g2 is continuous & g2 is one-to-one ) by A1, Th53;
A22: ( (g2 . O) `2 = 1 & (g2 . I) `1 = 1 ) by A1, Th55, Th56;
A23: ( - 1 <= (g2 . O) `1 & (g2 . O) `1 <= 1 & - 1 <= (g2 . I) `2 & (g2 . I) `2 <= 1 ) by A1, A4, A7, Th63;
X: rng g2 c= closed_inside_of_rectangle (- 1),1,(- 1),1 by A1, Th52;
then f2 . O,g2 . O,f2 . I,g2 . I are_in_this_order_on rectangle (- 1),1,(- 1),1 by A9, A10, A13, A14, A15, A16, A17, A18, A19, A20, A21, A22, A23, Th37;
then rng f2 meets rng g2 by A9, A10, A13, A14, A15, A16, A17, A18, A19, A20, A21, A22, A23, JGRAPH_6:89, X;
then A24: (rng f2) /\ (rng g2) <> {} by XBOOLE_0:def 7;
consider x being Element of (rng f2) /\ (rng g2);
A25: ( x in rng f2 & x in rng g2 ) by A24, XBOOLE_0:def 4;
then consider z1 being set such that
A26: ( z1 in dom f2 & x = f2 . z1 ) by FUNCT_1:def 5;
consider z2 being set such that
A27: ( z2 in dom g2 & x = g2 . z2 ) by A25, FUNCT_1:def 5;
A28: x = (AffineMap (2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) . (f . z1) by A11, A26, FUNCT_1:23;
A29: x = (AffineMap (2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) . (g . z2) by A12, A27, FUNCT_1:23;
A30: f . z1 in rng f by A11, A26, FUNCT_1:def 5;
A31: g . z2 in rng g by A12, A27, FUNCT_1:def 5;
f . z1 in the carrier of (TOP-REAL 2) by A26, FUNCT_2:7;
then A32: 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 A27, FUNCT_2:7;
then A33: 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 A8, TOPS_2:def 5;
then f . z1 = g . z2 by A28, A29, A32, A33, FUNCT_1:def 8;
hence rng f meets rng g by A30, A31, XBOOLE_0:3; :: thesis: verum