let P be non empty Subset of (TOP-REAL 2); :: thesis: for p1, p2, q1, q2 being Point of (TOP-REAL 2)

for g being Function of I[01],(TOP-REAL 2)

for s1, s2 being Real st g is continuous & g is one-to-one & rng g = P & g . 0 = p1 & g . 1 = p2 & g . s1 = q1 & 0 <= s1 & s1 <= 1 & g . s2 = q2 & 0 <= s2 & s2 <= 1 & LE q1,q2,P,p1,p2 holds

s1 <= s2

let p1, p2, q1, q2 be Point of (TOP-REAL 2); :: thesis: for g being Function of I[01],(TOP-REAL 2)

for s1, s2 being Real st g is continuous & g is one-to-one & rng g = P & g . 0 = p1 & g . 1 = p2 & g . s1 = q1 & 0 <= s1 & s1 <= 1 & g . s2 = q2 & 0 <= s2 & s2 <= 1 & LE q1,q2,P,p1,p2 holds

s1 <= s2

let g be Function of I[01],(TOP-REAL 2); :: thesis: for s1, s2 being Real st g is continuous & g is one-to-one & rng g = P & g . 0 = p1 & g . 1 = p2 & g . s1 = q1 & 0 <= s1 & s1 <= 1 & g . s2 = q2 & 0 <= s2 & s2 <= 1 & LE q1,q2,P,p1,p2 holds

s1 <= s2

let s1, s2 be Real; :: thesis: ( g is continuous & g is one-to-one & rng g = P & g . 0 = p1 & g . 1 = p2 & g . s1 = q1 & 0 <= s1 & s1 <= 1 & g . s2 = q2 & 0 <= s2 & s2 <= 1 & LE q1,q2,P,p1,p2 implies s1 <= s2 )

assume ( g is continuous & g is one-to-one & rng g = P ) ; :: thesis: ( not g . 0 = p1 or not g . 1 = p2 or not g . s1 = q1 or not 0 <= s1 or not s1 <= 1 or not g . s2 = q2 or not 0 <= s2 or not s2 <= 1 or not LE q1,q2,P,p1,p2 or s1 <= s2 )

then ex f being Function of I[01],((TOP-REAL 2) | P) st

( f = g & f is being_homeomorphism ) by Th16;

hence ( not g . 0 = p1 or not g . 1 = p2 or not g . s1 = q1 or not 0 <= s1 or not s1 <= 1 or not g . s2 = q2 or not 0 <= s2 or not s2 <= 1 or not LE q1,q2,P,p1,p2 or s1 <= s2 ) by JORDAN5C:def 3; :: thesis: verum

for g being Function of I[01],(TOP-REAL 2)

for s1, s2 being Real st g is continuous & g is one-to-one & rng g = P & g . 0 = p1 & g . 1 = p2 & g . s1 = q1 & 0 <= s1 & s1 <= 1 & g . s2 = q2 & 0 <= s2 & s2 <= 1 & LE q1,q2,P,p1,p2 holds

s1 <= s2

let p1, p2, q1, q2 be Point of (TOP-REAL 2); :: thesis: for g being Function of I[01],(TOP-REAL 2)

for s1, s2 being Real st g is continuous & g is one-to-one & rng g = P & g . 0 = p1 & g . 1 = p2 & g . s1 = q1 & 0 <= s1 & s1 <= 1 & g . s2 = q2 & 0 <= s2 & s2 <= 1 & LE q1,q2,P,p1,p2 holds

s1 <= s2

let g be Function of I[01],(TOP-REAL 2); :: thesis: for s1, s2 being Real st g is continuous & g is one-to-one & rng g = P & g . 0 = p1 & g . 1 = p2 & g . s1 = q1 & 0 <= s1 & s1 <= 1 & g . s2 = q2 & 0 <= s2 & s2 <= 1 & LE q1,q2,P,p1,p2 holds

s1 <= s2

let s1, s2 be Real; :: thesis: ( g is continuous & g is one-to-one & rng g = P & g . 0 = p1 & g . 1 = p2 & g . s1 = q1 & 0 <= s1 & s1 <= 1 & g . s2 = q2 & 0 <= s2 & s2 <= 1 & LE q1,q2,P,p1,p2 implies s1 <= s2 )

assume ( g is continuous & g is one-to-one & rng g = P ) ; :: thesis: ( not g . 0 = p1 or not g . 1 = p2 or not g . s1 = q1 or not 0 <= s1 or not s1 <= 1 or not g . s2 = q2 or not 0 <= s2 or not s2 <= 1 or not LE q1,q2,P,p1,p2 or s1 <= s2 )

then ex f being Function of I[01],((TOP-REAL 2) | P) st

( f = g & f is being_homeomorphism ) by Th16;

hence ( not g . 0 = p1 or not g . 1 = p2 or not g . s1 = q1 or not 0 <= s1 or not s1 <= 1 or not g . s2 = q2 or not 0 <= s2 or not s2 <= 1 or not LE q1,q2,P,p1,p2 or s1 <= s2 ) by JORDAN5C:def 3; :: thesis: verum