let f, g be Function of I[01],(TOP-REAL 2); :: thesis: for a, b, c, d being Real
for O, I being Point of I[01] st O = 0 & I = 1 & f is continuous & f is one-to-one & g is continuous & g is one-to-one & a <> b & c <> d & (f . O) `1 = a & c <= (f . O) `2 & (f . O) `2 <= d & (f . I) `1 = b & c <= (f . I) `2 & (f . I) `2 <= d & (g . O) `2 = c & a <= (g . O) `1 & (g . O) `1 <= b & (g . I) `2 = d & a <= (g . I) `1 & (g . I) `1 <= b & ( for r being Point of I[01] holds
( ( a >= (f . r) `1 or (f . r) `1 >= b or c >= (f . r) `2 or (f . r) `2 >= d ) & ( a >= (g . r) `1 or (g . r) `1 >= b or c >= (g . r) `2 or (g . r) `2 >= d ) ) ) holds
rng f meets rng g
let a, b, c, d be Real; :: thesis: for O, I being Point of I[01] st O = 0 & I = 1 & f is continuous & f is one-to-one & g is continuous & g is one-to-one & a <> b & c <> d & (f . O) `1 = a & c <= (f . O) `2 & (f . O) `2 <= d & (f . I) `1 = b & c <= (f . I) `2 & (f . I) `2 <= d & (g . O) `2 = c & a <= (g . O) `1 & (g . O) `1 <= b & (g . I) `2 = d & a <= (g . I) `1 & (g . I) `1 <= b & ( for r being Point of I[01] holds
( ( a >= (f . r) `1 or (f . r) `1 >= b or c >= (f . r) `2 or (f . r) `2 >= d ) & ( a >= (g . r) `1 or (g . r) `1 >= b or c >= (g . r) `2 or (g . r) `2 >= d ) ) ) holds
rng f meets rng g
let O, I be Point of I[01]; :: thesis: ( O = 0 & I = 1 & f is continuous & f is one-to-one & g is continuous & g is one-to-one & a <> b & c <> d & (f . O) `1 = a & c <= (f . O) `2 & (f . O) `2 <= d & (f . I) `1 = b & c <= (f . I) `2 & (f . I) `2 <= d & (g . O) `2 = c & a <= (g . O) `1 & (g . O) `1 <= b & (g . I) `2 = d & a <= (g . I) `1 & (g . I) `1 <= b & ( for r being Point of I[01] holds
( ( a >= (f . r) `1 or (f . r) `1 >= b or c >= (f . r) `2 or (f . r) `2 >= d ) & ( a >= (g . r) `1 or (g . r) `1 >= b or c >= (g . r) `2 or (g . r) `2 >= d ) ) ) implies rng f meets rng g )
assume that
A1: ( O = 0 & I = 1 & f is continuous & f is one-to-one & g is continuous & g is one-to-one ) and
A2: a <> b and
A3: c <> d and
A4: (f . O) `1 = a and
A5: ( c <= (f . O) `2 & (f . O) `2 <= d ) and
A6: ( (f . I) `1 = b & c <= (f . I) `2 & (f . I) `2 <= d & (g . O) `2 = c ) and
A7: ( a <= (g . O) `1 & (g . O) `1 <= b ) and
A8: ( (g . I) `2 = d & a <= (g . I) `1 & (g . I) `1 <= b & ( for r being Point of I[01] holds
( ( a >= (f . r) `1 or (f . r) `1 >= b or c >= (f . r) `2 or (f . r) `2 >= d ) & ( a >= (g . r) `1 or (g . r) `1 >= b or c >= (g . r) `2 or (g . r) `2 >= d ) ) ) ) ; :: thesis: rng f meets rng g
c <= d by A5, XXREAL_0:2;
then A9: c < d by A3, XXREAL_0:1;
a <= b by A7, XXREAL_0:2;
then a < b by A2, XXREAL_0:1;
hence rng f meets rng g by A1, A4, A5, A6, A7, A8, A9, JGRAPH_2:45; :: thesis: verum