reconsider O = 0 , I = 1 as Point of I[01] by BORSUK_1:40, XXREAL_1:1;
let p1, p2, p3, p4 be Point of (TOP-REAL 2); :: thesis: for a, b, c, d being Real
for f, g being Function of I[01],(TOP-REAL 2) st a < b & c < d & p1 `1 = b & p2 `1 = b & p3 `2 = c & p4 `2 = c & d >= p1 `2 & p1 `2 > p2 `2 & p2 `2 >= c & b >= p3 `1 & p3 `1 > p4 `1 & p4 `1 > a & 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; :: thesis: for f, g being Function of I[01],(TOP-REAL 2) st a < b & c < d & p1 `1 = b & p2 `1 = b & p3 `2 = c & p4 `2 = c & d >= p1 `2 & p1 `2 > p2 `2 & p2 `2 >= c & b >= p3 `1 & p3 `1 > p4 `1 & p4 `1 > a & 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 `1 = b & p2 `1 = b & p3 `2 = c & p4 `2 = c & d >= p1 `2 & p1 `2 > p2 `2 & p2 `2 >= c & b >= p3 `1 & p3 `1 > p4 `1 & p4 `1 > a & 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 that
A1: a < b and
A2: c < d and
A3: p1 `1 = b and
A4: p2 `1 = b and
A5: p3 `2 = c and
A6: p4 `2 = c and
A7: d >= p1 `2 and
A8: p1 `2 > p2 `2 and
A9: p2 `2 >= c and
A10: b >= p3 `1 and
A11: p3 `1 > p4 `1 and
A12: p4 `1 > a and
A13: f . 0 = p1 and
A14: f . 1 = p3 and
A15: g . 0 = p2 and
A16: g . 1 = p4 and
A17: ( f is continuous & f is one-to-one ) and
A18: ( g is continuous & g is one-to-one ) and
A19: rng f c= closed_inside_of_rectangle (a,b,c,d) and
A20: 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));
set h = AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))));
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) ;
A21: ( g2 is continuous & g2 is one-to-one ) by A1, A2, A18, Th53;
d > p2 `2 by A7, A8, XXREAL_0:2;
then A22: - 1 <= (g2 . O) `2 by A1, A2, A4, A9, A15, Th66;
A23: rng g2 c= closed_inside_of_rectangle ((- 1),1,(- 1),1) by A1, A2, A20, Th52;
A24: (g2 . I) `2 = - 1 by A2, A6, A16, Th57;
d - c > 0 by A2, XREAL_1:50;
then A25: 2 / (d - c) > 0 by XREAL_1:139;
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) ;
A26: ( f2 is continuous & f2 is one-to-one ) by A1, A2, A17, Th53;
p3 `1 > a by A11, A12, XXREAL_0:2;
then A27: (f2 . I) `1 <= 1 by A1, A2, A5, A10, A14, Th66;
A28: rng f2 c= closed_inside_of_rectangle ((- 1),1,(- 1),1) by A1, A2, A19, Th52;
A29: (f2 . I) `2 = - 1 by A2, A5, A14, Th57;
p1 `2 >= c by A8, A9, XXREAL_0:2;
then A30: (f2 . O) `2 <= 1 by A1, A2, A3, A7, A13, Th66;
A31: (f2 . O) `1 = 1 by A1, A3, A13, Th56;
set x = the Element of (rng f2) /\ (rng g2);
A32: dom f = the carrier of I[01] by FUNCT_2:def 1;
then A33: (AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p3 = f2 . I by A14, FUNCT_1:13;
b - a > 0 by A1, XREAL_1:50;
then A34: 2 / (b - a) > 0 by XREAL_1:139;
then A35: ((AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p1) `2 > ((AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p2) `2 by A8, A25, Th51;
b > p4 `1 by A10, A11, XXREAL_0:2;
then A36: - 1 < (g2 . I) `1 by A1, A2, A6, A12, A16, Th66;
AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) is being_homeomorphism by A34, A25, Th51;
then A37: AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c)))) is one-to-one by TOPS_2:def 5;
A38: (g2 . O) `1 = 1 by A1, A4, A15, Th56;
A39: dom g = the carrier of I[01] by FUNCT_2:def 1;
then A40: (AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p2 = g2 . O by A15, FUNCT_1:13;
A41: ((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 A11, A34, A25, Th50;
A42: (AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p4 = g2 . I by A16, A39, FUNCT_1:13;
(AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . p1 = f2 . O by A13, A32, FUNCT_1:13;
then f2 . O,g2 . O,f2 . I,g2 . I are_in_this_order_on rectangle ((- 1),1,(- 1),1) by A35, A41, A40, A33, A42, A31, A29, A30, A27, A38, A24, A22, A36, Th46;
then rng f2 meets rng g2 by A26, A28, A21, A23, JGRAPH_6:79;
then A43: (rng f2) /\ (rng g2) <> {} by XBOOLE_0:def 7;
then the Element of (rng f2) /\ (rng g2) in rng g2 by XBOOLE_0:def 4;
then consider z2 being object such that
A44: z2 in dom g2 and
A45: the Element of (rng f2) /\ (rng g2) = g2 . z2 by FUNCT_1:def 3;
A46: the Element of (rng f2) /\ (rng g2) = (AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . (g . z2) by A39, A44, A45, FUNCT_1:13;
A47: g . z2 in rng g by A39, A44, FUNCT_1:def 3;
g . z2 in the carrier of (TOP-REAL 2) by A44, FUNCT_2:5;
then A48: g . z2 in dom (AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) by FUNCT_2:def 1;
the Element of (rng f2) /\ (rng g2) in rng f2 by A43, XBOOLE_0:def 4;
then consider z1 being object such that
A49: z1 in dom f2 and
A50: the Element of (rng f2) /\ (rng g2) = f2 . z1 by FUNCT_1:def 3;
A51: f . z1 in rng f by A32, A49, FUNCT_1:def 3;
f . z1 in the carrier of (TOP-REAL 2) by A49, FUNCT_2:5;
then A52: f . z1 in dom (AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) by FUNCT_2:def 1;
the Element of (rng f2) /\ (rng g2) = (AffineMap ((2 / (b - a)),(- ((b + a) / (b - a))),(2 / (d - c)),(- ((d + c) / (d - c))))) . (f . z1) by A32, A49, A50, FUNCT_1:13;
then f . z1 = g . z2 by A46, A52, A48, A37, FUNCT_1:def 4;
hence rng f meets rng g by A51, A47, XBOOLE_0:3; :: thesis: verum