let a, b, c, d be real number ; :: thesis: for f, g being continuous Function of I[01] ,(TOP-REAL 2)
for O, I being Point of I[01] st O = 0 & I = 1 & (f . O) `1 = a & (f . I) `1 = b & (g . O) `2 = c & (g . I) `2 = d & ( for r being Point of I[01] holds
( a <= (f . r) `1 & (f . r) `1 <= b & a <= (g . r) `1 & (g . r) `1 <= b & c <= (f . r) `2 & (f . r) `2 <= d & c <= (g . r) `2 & (g . r) `2 <= d ) ) holds
rng f meets rng g
let f, g be continuous Function of I[01] ,(TOP-REAL 2); :: thesis: for O, I being Point of I[01] st O = 0 & I = 1 & (f . O) `1 = a & (f . I) `1 = b & (g . O) `2 = c & (g . I) `2 = d & ( for r being Point of I[01] holds
( a <= (f . r) `1 & (f . r) `1 <= b & a <= (g . r) `1 & (g . r) `1 <= b & c <= (f . r) `2 & (f . r) `2 <= d & c <= (g . r) `2 & (g . r) `2 <= d ) ) holds
rng f meets rng g
let O, I be Point of I[01] ; :: thesis: ( O = 0 & I = 1 & (f . O) `1 = a & (f . I) `1 = b & (g . O) `2 = c & (g . I) `2 = d & ( for r being Point of I[01] holds
( a <= (f . r) `1 & (f . r) `1 <= b & a <= (g . r) `1 & (g . r) `1 <= b & c <= (f . r) `2 & (f . r) `2 <= d & c <= (g . r) `2 & (g . r) `2 <= d ) ) implies rng f meets rng g )
assume A1: ( O = 0 & I = 1 & (f . O) `1 = a & (f . I) `1 = b & (g . O) `2 = c & (g . I) `2 = d & ( for r being Point of I[01] holds
( a <= (f . r) `1 & (f . r) `1 <= b & a <= (g . r) `1 & (g . r) `1 <= b & c <= (f . r) `2 & (f . r) `2 <= d & c <= (g . r) `2 & (g . r) `2 <= d ) ) ) ; :: thesis: rng f meets rng g
reconsider P = rng f as non empty Subset of (TOP-REAL 2) ;
reconsider Q = rng g as non empty Subset of (TOP-REAL 2) ;
A2: dom f = the carrier of I[01] by FUNCT_2:def 1;
A3: dom g = the carrier of I[01] by FUNCT_2:def 1;
the carrier of ((TOP-REAL 2) | P) = [#] ((TOP-REAL 2) | P)
.= rng f by PRE_TOPC:def 10 ;
then reconsider f1 = f as Function of I[01] ,((TOP-REAL 2) | P) by A2, FUNCT_2:3;
the carrier of ((TOP-REAL 2) | Q) = [#] ((TOP-REAL 2) | Q)
.= rng g by PRE_TOPC:def 10 ;
then reconsider g1 = g as Function of I[01] ,((TOP-REAL 2) | Q) by A3, FUNCT_2:3;
reconsider p1 = f1 . O as Point of (TOP-REAL 2) by A1;
reconsider p2 = f1 . I as Point of (TOP-REAL 2) by A1;
reconsider q1 = g1 . O as Point of (TOP-REAL 2) by A1;
reconsider q2 = g1 . I as Point of (TOP-REAL 2) by A1;
A4: ( f is Path of p1,p2 & g is Path of q1,q2 ) by A1, BORSUK_2:def 4;
A5: for p being Point of (TOP-REAL 2) st p in P holds
( p1 `1 <= p `1 & p `1 <= p2 `1 )

proof

let p be Point of (TOP-REAL 2); :: thesis: ( p in P implies ( p1 `1 <= p `1 & p `1 <= p2 `1 ) )
assume p in P ; :: thesis: ( p1 `1 <= p `1 & p `1 <= p2 `1 )
then ex x being set st
( x in dom f1 & p = f1 . x ) by FUNCT_1:def 5;
hence ( p1 `1 <= p `1 & p `1 <= p2 `1 ) by A1; :: thesis: verum

end;

A6: for p being Point of (TOP-REAL 2) st p in Q holds
( p1 `1 <= p `1 & p `1 <= p2 `1 )

proof

let p be Point of (TOP-REAL 2); :: thesis: ( p in Q implies ( p1 `1 <= p `1 & p `1 <= p2 `1 ) )
assume p in Q ; :: thesis: ( p1 `1 <= p `1 & p `1 <= p2 `1 )
then ex x being set st
( x in dom g1 & p = g1 . x ) by FUNCT_1:def 5;
hence ( p1 `1 <= p `1 & p `1 <= p2 `1 ) by A1; :: thesis: verum

end;

A7: for p being Point of (TOP-REAL 2) st p in P holds
( q1 `2 <= p `2 & p `2 <= q2 `2 )

proof

let p be Point of (TOP-REAL 2); :: thesis: ( p in P implies ( q1 `2 <= p `2 & p `2 <= q2 `2 ) )
assume p in P ; :: thesis: ( q1 `2 <= p `2 & p `2 <= q2 `2 )
then ex x being set st
( x in dom f1 & p = f1 . x ) by FUNCT_1:def 5;
hence ( q1 `2 <= p `2 & p `2 <= q2 `2 ) by A1; :: thesis: verum

end;

for p being Point of (TOP-REAL 2) st p in Q holds
( q1 `2 <= p `2 & p `2 <= q2 `2 )

proof

let p be Point of (TOP-REAL 2); :: thesis: ( p in Q implies ( q1 `2 <= p `2 & p `2 <= q2 `2 ) )
assume p in Q ; :: thesis: ( q1 `2 <= p `2 & p `2 <= q2 `2 )
then ex x being set st
( x in dom g1 & p = g1 . x ) by FUNCT_1:def 5;
hence ( q1 `2 <= p `2 & p `2 <= q2 `2 ) by A1; :: thesis: verum

end;

hence rng f meets rng g by A4, A5, A6, A7, Th4; :: thesis: verum