let IT1, IT2 be Point of (TOP-REAL 2); ( IT1 in P /\ Q & ( for g being Function of I[01],((TOP-REAL 2) | P)
for s2 being Real st g is being_homeomorphism & g . 0 = p1 & g . 1 = p2 & g . s2 = IT1 & 0 <= s2 & s2 <= 1 holds
for t being Real st 1 >= t & t > s2 holds
not g . t in Q ) & IT2 in P /\ Q & ( for g being Function of I[01],((TOP-REAL 2) | P)
for s2 being Real st g is being_homeomorphism & g . 0 = p1 & g . 1 = p2 & g . s2 = IT2 & 0 <= s2 & s2 <= 1 holds
for t being Real st 1 >= t & t > s2 holds
not g . t in Q ) implies IT1 = IT2 )
A5:
P /\ Q c= P
by XBOOLE_1:17;
assume that
A6:
IT1 in P /\ Q
and
A7:
for g1 being Function of I[01],((TOP-REAL 2) | P)
for s1 being Real st g1 is being_homeomorphism & g1 . 0 = p1 & g1 . 1 = p2 & g1 . s1 = IT1 & 0 <= s1 & s1 <= 1 holds
for t being Real st 1 >= t & t > s1 holds
not g1 . t in Q
; ( not IT2 in P /\ Q or ex g being Function of I[01],((TOP-REAL 2) | P) ex s2 being Real st
( g is being_homeomorphism & g . 0 = p1 & g . 1 = p2 & g . s2 = IT2 & 0 <= s2 & s2 <= 1 & ex t being Real st
( 1 >= t & t > s2 & g . t in Q ) ) or IT1 = IT2 )
assume that
A8:
IT2 in P /\ Q
and
A9:
for g2 being Function of I[01],((TOP-REAL 2) | P)
for s2 being Real st g2 is being_homeomorphism & g2 . 0 = p1 & g2 . 1 = p2 & g2 . s2 = IT2 & 0 <= s2 & s2 <= 1 holds
for t being Real st 1 >= t & t > s2 holds
not g2 . t in Q
; IT1 = IT2
consider g being Function of I[01],((TOP-REAL 2) | P) such that
A10:
g is being_homeomorphism
and
A11:
( g . 0 = p1 & g . 1 = p2 )
by A2, TOPREAL1:def 1;
A12: rng g =
[#] ((TOP-REAL 2) | P)
by A10, TOPS_2:def 5
.=
P
by PRE_TOPC:def 5
;
then consider ss1 being object such that
A13:
ss1 in dom g
and
A14:
g . ss1 = IT1
by A5, A6, FUNCT_1:def 3;
reconsider ss1 = ss1 as Real by A13, BORSUK_1:40;
A15:
0 <= ss1
by A13, BORSUK_1:43;
A16:
( P /\ Q c= Q & ss1 <= 1 )
by A13, BORSUK_1:43, XBOOLE_1:17;
consider ss2 being object such that
A17:
ss2 in dom g
and
A18:
g . ss2 = IT2
by A5, A8, A12, FUNCT_1:def 3;
reconsider ss2 = ss2 as Real by A17, BORSUK_1:40;
A19:
0 <= ss2
by A17, BORSUK_1:43;
A20:
ss2 <= 1
by A17, BORSUK_1:43;