let P be Subset of (); :: thesis: for p1, p2, q1, q2, q3 being Point of () st LE q1,q2,P,p1,p2 & LE q2,q3,P,p1,p2 holds
LE q1,q3,P,p1,p2

let p1, p2, q1, q2, q3 be Point of (); :: thesis: ( LE q1,q2,P,p1,p2 & LE q2,q3,P,p1,p2 implies LE q1,q3,P,p1,p2 )
assume that
A1: LE q1,q2,P,p1,p2 and
A2: LE q2,q3,P,p1,p2 ; :: thesis: LE q1,q3,P,p1,p2
A3: q2 in P by A1;
A4: now :: thesis: for g being Function of I,(() | P)
for s1, s2 being Real st g is being_homeomorphism & g . 0 = p1 & g . 1 = p2 & g . s1 = q1 & 0 <= s1 & s1 <= 1 & g . s2 = q3 & 0 <= s2 & s2 <= 1 holds
s1 <= s2
A5: [.0,1.] = { r1 where r1 is Real : ( 0 <= r1 & r1 <= 1 ) } by RCOMP_1:def 1;
let g be Function of I,(() | P); :: thesis: for s1, s2 being Real st g is being_homeomorphism & g . 0 = p1 & g . 1 = p2 & g . s1 = q1 & 0 <= s1 & s1 <= 1 & g . s2 = q3 & 0 <= s2 & s2 <= 1 holds
s1 <= s2

let s1, s2 be Real; :: thesis: ( g is being_homeomorphism & g . 0 = p1 & g . 1 = p2 & g . s1 = q1 & 0 <= s1 & s1 <= 1 & g . s2 = q3 & 0 <= s2 & s2 <= 1 implies s1 <= s2 )
assume that
A6: g is being_homeomorphism and
A7: ( g . 0 = p1 & g . 1 = p2 & g . s1 = q1 ) and
0 <= s1 and
A8: ( s1 <= 1 & g . s2 = q3 & 0 <= s2 & s2 <= 1 ) ; :: thesis: s1 <= s2
rng g = [#] (() | P) by
.= P by PRE_TOPC:def 5 ;
then consider x being object such that
A9: x in dom g and
A10: q2 = g . x by ;
dom g = [#] I by
.= the carrier of I ;
then consider s3 being Real such that
A11: ( s3 = x & 0 <= s3 & s3 <= 1 ) by ;
( s1 <= s3 & s3 <= s2 ) by A1, A2, A6, A7, A8, A10, A11;
hence s1 <= s2 by XXREAL_0:2; :: thesis: verum
end;
( q1 in P & q3 in P ) by A1, A2;
hence LE q1,q3,P,p1,p2 by A4; :: thesis: verum